1.1 Admission

The details of admission are available in the prospectus. Students having the B.Stat. degree of ISI, B.A./B.Sc. (Statistics), B.Math. degree of ISI, or B.A./B.Sc. (Mathematics) with Statistics as a full subject are eligible for the M.Stat. selection test. Students with B.Stat. (Hons.) degree of ISI are eligible for direct admission to the M.Stat. programme. Any student who is asked to discontinue the M.Stat. programme is not eligible for readmission into this programme even through admission test.

1.2 Duration

The total duration of the M.Stat. programme is four semesters. An academic year usually starts in July-August and continues till May, consisting of two semesters with a recess in-between. There is a study-break of one week before the semestral examinations in each semester. All M. Stat. students are required to undergo a training in ``National and International Statistical Systems'' at the CSO, New Delhi at the end of their First Year (usually May--June).

1.3 Courses

The M.Stat. programme has two streams: B-stream for students having the B.Stat. degree and NB-stream for others. There are twenty courses for the B-stream and twenty-one courses for the NB-stream. The set of courses to be taken in a semester depends on the stream, choice of optional courses and the choice of specialization in the second year. The training in official statistics at the C.S.O., New Delhi, is a part of the M.Stat. programme.

1.4 Examinations

The final (semestral) examination in a course is held at the end of the semester. Besides, there is a mid-semestral examination in each course. The calendar for the semester is announced in advance. Usually, the scores of homeworks/assignments, mid-semestral and semestral examinations are combined to get the composite score in a course, as explained in Section 1.5 below. The mid-semestral examinations are held over a period of one week. This examination is of a shorter duration than the semestral examination, and should not last more than two hours.

If the composite score of a student falls short of 45% in a credit course, or 35% in a non-credit course, the student may take a back-paper examination to improve the score. At most one back-paper examination is allowed in each course. Moreover, a student can take at most four back-paper examinations (for credit courses) in the first year and only two in the second year. The decision to allow a student to appear for the back-paper examination is taken by the appropriate Teachers' Committee. The back-paper examination covers the entire syllabus of the course.

If a student misses the mid-semestral examination of a course due to medical or family emergency, he/she is allowed to take a supplementary mid-semestral examination. When a student fails to appear at the semestral examination of a course due to illness or other compelling reasons, the Teachers' Committee may, on an adequately documented representation from the student, allow him/her to take a supplementary examination in the course. This examination is held at the same time as the back-paper examinations for that semester and a student taking this examination is not given any other examination in the course. The maximum a student can score in a supplementary examination is 60%.

A student may take more than the allotted quota of backpaper examinations in a given academic year, and decide at the end of that academic year which of the BP exam scores should be disregarded.

1.5 Scores

The composite score in a course is a weighted average of the scores in the mid-semestral and semestral examinations, home-assignments, and/or project work in that course; the weights are announced beforehand by the Dean of Studies, or the Class Teacher, in consultation with the teacher concerned .

 The minimum composite score to pass a course is 35%.

 When a student takes back-paper examination in a credit course, his final score in that course is the higher of the back-paper score and the earlier composite score, subject to a maximum of 45%. When a student takes supplementary mid-semestral or semestral examination in a course, the maximum he/she can score in that examination is 60%. Unlike the back-paper examination, the score in the supplementary examination is used along with other scores to arrive at the composite score.

1.6 Attendance

Each student is required to attend at least 75% of all the classes held in each academic year, failing which he/she is not allowed to appear at the second semestral examination (leading to discontinuation from the programme).

Less than 75% attendance record in the first semester in any academic year leads to reduction of stipend in the following semester; see Section 1.10.

Students with attendance more than 50% but less than 75% in the first semester in any academic year is given warning and urged to improve their attendance.

If a student fails to attend classes in any course continuously for one week or more, he/she would be required to furnish explanation to the Dean of Studies or the Class Teacher for such absence. If such explanation is found to be satisfactory by the Teachers' Committee, then the calculation of percentage of attendance is determined disregarding the period for which explanation has been provided by the student and accepted by the Teachers' Committee. Stipend may be fully or partially withdrawn for a specific period by the respective Teachers' Committee if such explanations are not given, or if such explanations are found to be not satisfactory by the Teachers' Committee.--> If a teacher reports insufficient attendance on the part of any student, the Teachers' Committee/Dean of Studies may ask for explanation from the student. In case a satisfactory explanation is not available, the student may be warned and his/her guardian informed. A student is also required to furnish proper notice in time and provide satisfactory explanation if he/she fails to take any examination.

1.7 Promotion

Here and in what follows, copying in the examination, rowdyism or some other breach of discipline or unlawful/unethical behaviour etc. are regarded as unsatisfactory conduct.

A student is considered for promotion to the next year of the programme only when he/she meets the attendance requirement and his/her conduct has been satisfactory. Subject to the above conditions, a student is promoted from First Year to Second Year if the average composite score in all credit courses taken in the first year is not less than 45%, and no composite score in a course is less than 35%.

1.8 Final Result

At the end of the second year the overall average of the percentage composite scores in all the credit courses taken in the two-year programme is computed for each student. The student is awarded the M.Stat. degree in one of the following categories according to the criteria he/she satisfies, provided, in the second year, he/she does not have a composite score of less than 35% in a course, and his/her conduct is satisfactory.

Final Result


M.Stat., First Division with Distinction

The overall average score is at least 75% and the composite score in at most two credit courses is less than 45%.


First Division

(i) Not in First Division with Distinction, (ii) the overall average score is at least 60%, and (iii) the composite score in at most four credit courses is less than 45%.


Second Division

(i) Not in First Division with Distinction or First Division, (ii) the overall average score is at least 45%, and (iii) the composite score in at most four credit courses is less than 45%.


The students who fail but obtain at least 35% average score in the second year, and have satisfactory conduct are allowed to repeat the final year of the M.Stat. programme without stipend; the scores obtained during the repetition of the second year are taken as the final scores in the second year. A student is not given more than one chance to repeat the second year of the programme.

1.9 Class-Teacher

One of the instructors of a class is designated as the Class Teacher. Students are required to meet their respective Class Teachers periodically to get their academic performance reviewed, and to discuss their problems regarding courses.

1.10 Stipends

Stipend, if awarded at the time of admission, is valid initially for the first semester only. The amount of stipend to be awarded in each subsequent semester will depend on academic performance, conduct, and attendance, as specified below, provided the requirements for continuation of the academic programme (excluding repetition) are satisfied; see Sections 1.6 and 1.7.

    1. All composite scores used in the following are considered after the respective back-paper examinations.
    2. If all the requirements for continuation of the programme are satisfied, and the average composite score is at least 60% and the number of credit course scores less than 45% is at most one in any particular semester, the full value of the stipend is awarded in the following semester.
    3. If all the requirements for continuation of the programme are satisfied, and the average composite score is at least 45% and the number of credit course scores less than 45% is at most one in any particular semester, then half stipend is awarded in the following semester.
    4. In all cases other than (2) and (3) above, no stipend is awarded in the following semester.
    1. If the overall attendance in all courses in the first semester in any academic year is less than 75%, no stipend is awarded in the following semester.

The Dean of Studies or the Class Teacher, at any time, in consultation with the respective Teachers' Committee, may withdraw the stipend of a student fully for a specific period if his/her conduct in the campus is found to be unsatisfactory.

Note: The net amount of the stipend to be awarded is determined by simultaneous and concurrent application of all clauses described above; but, in no case, the amount of stipend to be awarded or to be withdrawn should exceed 100% of the prescribed amount of stipend.

Stipends can be restored because of improved performance and/or attendance, but no stipend is restored with retrospective effect.

Stipends are given after the end of each month for eleven months in each academic year. The first stipend is given two months after admission with retrospective effect provided the student continues in the M.Stat. programme for at least two months. Stipends are given to the M.Stat. students during their CSO training programme at New Delhi.

Contingency grants can be used for purchasing a scientific calculator and other required accessories for the practical class, text books and supplementary text books and for getting photostat copies of required academic material. All such expenditure should be approved by the Class Teacher. No contingency grants are given in the first two months after admission.

1.11 ISI Library Rules

Any student is allowed to use the reading room facilities in the library and allowed access to the stacks. M.Stat. students have to pay a security deposit of Rs. 100 in order to avail the borrowing facility. A student can borrow at most four books at a time.

Any book from the Text Book Library (TBL) collection may be issued out to a student only for overnight or week-end provided at least one copy of that book is left in the TBL. Only one TBL book is issued at a time to a student. Fine is charged if any book is not returned by the due date stamped on the issue-slip. The library rules, and other details are posted in the library.

1.12 Change of Rules

The Institute reserves the right to make changes in the above rules, course structure and the syllabi as and when needed.


The M.Stat. programme is of two years' duration. Two different curricula are offered in the first year of this programme - one for the NB-Stream and the other for the B-Stream. Students have to choose one of the following specializations in the second year :

Offering a specialization is subject to availability of students and adequate resources and not all specializations may be given at all centres. Each specialization has a number of prerequisites in terms of specific courses. The maximum class size of any particular specialization in a centre is also limited. The final selection of students for various specializations is determined by the Dean of Studies in consultation with the Teachers' Committee. All the courses listed below are allocated three or four lecture sessions and one practical-cum-tutorial session per week. The practical-cum-tutorial session consists of one or two periods. These periods are meant to be used for discussion on problems, practicals, discussion of computer outputs, assignments, for special lectures and self-study, etc. All these need not be contact hours.

First Year Curriculum


First Year, First Semester






Linear Algebra and Linear Models


Large Sample Statistical Methods


Real Analysis


Measure Theoretic Probability


Sample Surveys and Design of Experiments


Sample Surveys and Design of Experiments


Large Sample Theory and Markov Chain


Applied Stochastic Processes


Statistical Inference I


Statistical Inference I

First Year, Second Semester






Regression Techniques


Regression Techniques


Multivariate Analysis


Multivariate Analysis


Measure Theoretic Probability


Metric Topology and Complex Analysis


Elective Course I


Elective Course I


Elective Course II


Elective Course II


Programming and Data Structures



List of First Year Second Semester Elective Courses


Time Series Analysis (required for AS, ASDA, BSDA and QE specializations)


Optimization Techniques (required for ISOR specialization)


Metric Topology and Complex Analysis (required for MSP and AP, available only to NB-stream students)


Nonparametric and Sequential Analysis (available only to NB-stream students)


Discrete Mathematics


Training course on `National and International Statistical Systems: It is a non-credit course offered at the end of the first year in collaboration with the Central Statistical Organization, New Delhi. In case of failure in the course, even after the back-paper examination, a student may be allowed, in exceptional cases, to undergo training for a second time at his/her expense at the end of the second year.


Second Year Curriculum for Various Specializations

The specializations to be offered at different centres are announced beforehand. Each student is asked to give his/her options for different specializations. Each student will be admitted to a particular specialization based on his/her options and academic background. A student has to take ten courses in the second year, out of which a specified number of courses has to be taken from the selected specialization; the other courses may be taken from the entire list of courses offered in the second year. The courses to be offered in each semester for different specializations are announced beforehand depending on availability of resources.

Advanced Probability (AP) Specialization

A student in the AP specialization has to take four compulsory courses and six elective courses, at least three of which must be from the Main List of Elective Courses for AP. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Advanced Probability I
  2. Functional Analysis
  3. Elective I
  4. Elective II
  5. Elective III

Second  Semester

  1. Stochastic Processes I
  2. Stochastic Processes II
  3. Elective IV
  4. Elective V
  5. Elective VI

Main List of Elective Courses (at least THREE courses from this list)

  1. Ordinary and Partial Differential Equations
  2. Advanced Functional Analysis
  3. Ergodic Theory
  4. Quantum Probability
  5. Stochastic Integration
  6. Multidimensional Diffusions
  7. Theory of Large Deviations
  8. Probability on Banach Spaces
  9. Statistical Mechanics
  10. Inference in Stochastic Processes
  11. Topics in Set Theory
  12. Topics in Fourier Analysis
  13. Linear Lie Groups and Their Representations
  14. Calculus on Manifolds
  15. Percolation Theory
  16. Martingale Problems and Markov Processes

Supplementary List of Elective Courses

  1. Special Topics in AP

Actuarial Statistics (AS) Specialization

A student in the AS specialization has to take four compulsory courses including a project and six elective courses, at least four of which must be from the Main List of Elective Courses for AS. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Actuarial Methods
  2. Life Contingencies
  3. Elective I
  4. Elective II
  5. Elective III

Second  Semester

  1. Actuarial Models
  2. Survival Analysis
  3. Elective IV
  4. Elective V
  5. Elective VI

Main List of Elective Courses (at least FOUR courses from this list)

  1. Statistical Computing
  2. Game Theory I
  3. Microeconomic Theory I
  4. Macroeconomic Theory I
  5. Theory of Finance I
  6. Theory of Finance II
  7. Applied Multivariate Analysis
  8. Life Testing and Reliability
  9. Theory of Games and Statistical Decisions
  10. Stochastic Processes I
  11. Econometric Methods
  12. Statistical Methods in Demography

Supplementary List of Elective Courses

  1. Special Topics in AS

Applied Statistics and Data Analysis (ASDA) Specialization

A student in the ASDA specialization has to take six compulsory courses including a project and four elective courses, at least two of which must be from the Main List of  Elective Courses for ASDA. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Advanced Design of Experiments
  2. Analysis of Discrete Data
  3. Statistical Computing
  4. Elective I
  5. Elective II

Second  Semester

  1. Advanced Sample Surveys
  2. Applied Multivariate Analysis
  3. Elective III
  4. Elective IV
  5. Project

Main List of Elective Courses (at least TWO courses from this list)

  1. Survival Analysis
  2. Statistical Methods in Genetics I
  3. Biostatistics
  4. Life Testing and Reliability
  5. Theory of Games and Statistical Decisions
  6. Econometric Methods
  7. Quantitative Models in Social Sciences
  8. Pattern Recognition and Image Processing

Supplementary List of Elective Courses

  1. Special Topics in ASDA

Biostatistics and Data Analysis (BSDA) Specialization

A student in the BSDA specialization has to take seven compulsory courses including a project and three elective courses, at least one of which must be from the Main List of  Elective Courses for BSDA. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Statistical Methods in Genetics I
  2. Analysis of Discrete Data
  3. Statistical Computing
  4. Elective I
  5. Elective II

Second  Semester

  1. Survival Analysis
  2. Statistical Methods in Public Health
  3. Statistical Methods in Biomedical Research
  4. Elective III
  5. Project

Main List of Elective Courses (at least ONE courses from this list)

  1. Advanced Design of Experiments
  2. Advanced Sample Surveys
  3. Applied Multivariate Analysis
  4. Life Testing and Reliability
  5. Theory of Games and Statistical Decisions
  6. Statistical Ecology
  7. Statistical Methods in Genetics II
  8. Statistical Methods in Demography
  9. Pattern Recognition and Image Processing

Supplementary List of Elective Courses

  1. Special Topics in BSDA

Industrial Statistics and Operations Research (ISOR) Specialization

A student in the ISOR specialization has to take seven compulsory courses including a project and three elective courses, which can be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Advanced Design of Experiments
  2. Life Testing and Reliability
  3. Quality Control and Its Management
  4. Elective I
  5. Elective II

Second  Semester

  1. Management Applications of Optimization
  2. Industrial Applications of Stochastic Processes
  3. Optimization Techniques II
  4. Elective III
  5. Project

There is NO Main List of Elective Courses for ISOR Specialization

Supplementary List of Elective Courses

  1. Optimization Techniques III
  2. Optimization Techniques IV
  3. Network Analysis
  4. Sampling Inspection
  5. Scheduling Theory
  6. Industrial Engineering and Management
  7. Special Topics in ISOR

Mathematical Statistics and Probability (MSP) Specialization

A student in the MSP specialization has to take four compulsory courses and six elective courses, at least three of which must be from the Main List of Elective Courses for MSP. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year. An MSP student with the B.Stat. (Hons) background can opt for a two-semester dissertation in lieu of two Main Elective courses provided s/he has the average of at least 80% marks in all the Statistics and Probability courses of the B.Stat. (Hons.) curriculum and at least 85% marks in the first year of the M.Stat. curriculum.

First Semester

  1. Advanced Probability I
  2. Functional Analysis
  3. Elective I
  4. Elective II
  5. Elective III

Second  Semester

  1. Stochastic Processes I
  2. Statistical Inference II
  3. Elective IV
  4. Elective V
  5. Elective VI

Main List of Elective Courses (at least THREE courses from this list)

  1. Nonparametric Inference
  2. Advanced Design of Experiments
  3. Advanced Sample Surveys
  4. Theory of Games and Statistical Decisions
  5. Sequential Analysis and Optimal Stopping
  6. Topics in Bayesian Inference
  7. Time Series Analysis
  8. Asymptotic Theory of Inference
  9. Pattern Recognition and Image Processing
  10. Statistical Computing
  11. Dissertation

Supplementary List of Elective Courses

  1. Topology and Set Theory
  2. Graph Theory and Combinatorics
  3. Advanced Algebra
  4. Harmonic Analysis
  5. Topics in Mathematical Logic
  6. Algebraic Topology
  7. Application of Analysis to Geometry
  8. Descriptive Set Theory
  9. Advanced Probability II
  10. Second-order Processes
  11. Topics in Mathematical Physics
  12. Special Topics in MSP

Quantitative Economics (QE) Specialization

This specialization is available to students with B.Stat. background who have had Group I (Economics) elective courses, and to students with B.Sc. (Stat/Math) background who have had Economics as a full subject. A student in the QE specialization has to take four compulsory courses and six elective courses, at least four of which must be from the Main List of  Elective Courses for QE. The remaining elective courses may be chosen from any of the M.Stat. courses offered in the second year.

First Semester

  1. Microeconomic Theory I
  2. Game Theory I
  3. Econometric Methods
  4. Elective I
  5. Elective II

Second  Semester

  1. Macroeconomic Theory I
  2. Elective III
  3. Elective IV
  4. Elective V
  5. Elective VI

Main List of Elective Courses (at least FOUR courses from this list)

  1. Microeconomic Theory II
  2. Macroeconomic Theory II
  3. Agricultural Economics
  4. Industrial Organization
  5. Economic Development I
  6. Modern Growth Theory
  7. Social Choice and Political Economy
  8. Incentives and Organizations
  9. Privatization and Regulation
  10. Economic Development II
  11. Econometric Applications II
  12. Game Theory II
  13. Bayesian Econometrics
  14. Intertemporal Economics
  15. Theory of Planning
  16. Social Accounting
  17. Public Economics
  18. Regional Economics
  19. International Economics I
  20. International Economics II
  21. Advanced Topics in International Economics
  22. Mathematical Programming with Applications to Economics
  23. Monetary Economics
  24. History of Economic Thought
  25. Environmental Economics
  26. Theory of Finance I
  27. Theory of Finance II
  28. Theory of Finance III
  29. Political Economy and Comparative Systems

Supplementary List of Elective Courses

  1. Special Topics in QE


(Summary of the proposed syllabi)


The syllabi given for the first year courses and the second year compulsory courses should be adhered to by the instructor as much as possible. The syllabi for the second year elective courses are meant to serve as guidelines.

First Year Courses

Real Analysis (for NB-stream only - this course is meant to prepare students for Measure Theoretic Probability, to be taught in the second semester)
A quick review of real number system, open/closed sets sequences and series, continuous functions, mean value theorem and Taylor expansions. Riemann integral, sequences of functions, uniform convergence, power series, continuity and differentiability in several variables.  [Roughly as in T.M. Apostol's Mathematical Analysis]

Metric Topology and Complex Analysis
  Metric spaces, open/closed sets, compactness, completeness, Baire category theorem, connectedness, continuous functions and homeomorphisms. Product spaces, C[0,1] and L^p spaces as examples of complete spaces. (about 1/3 time to be spent)
Analytic functions, Cauchy-Riemann equations, Cauchy/Morera theorems, Cauchy integral formula, Liouville's theorem, singularities and Cauchy residual formula, contour integration, Rouche's theorem, Fractional linear transformations (about 2/3 time to be spent)
  [Roughly the relevant parts from `Introduction to Topology and Modern Analysis' by G.F. Simmons and `Real and Complex Analysis' by W. Rudin]

Measure Theoretic Probability
Measure and integration: \sigma fields and monotone class theorem, probability measures, statement of Caratheodory extension theorem, measurable functions, integration, Fatou, MCT, DCT, product spaces, Fubini. (about 1/2 time to be spent)
Probability: 1-1 correspondence between distribution functions and probabilities on R, independence, Borel-Cantelli, Weak and Strong laws in the i.i.d. case, Kolmogorov 0-1 law, various modes of convergence, characterstic functions, uniqueness/inversion/Levy continuity theorems, proof of CLT for the iid case with finite variance. (about 1/2 time to be spent)
[Roughly as in `Probability and Measure' by P. Billingsley]
[The order of coverage depends on teacher. For instance 1-1 correspondence can come soon after Caratheodory]

Large Sample Theory and Markov Chain (for NB-stream only)
(1) Review of various modes of convergence of random variables and Central Limit Theorems. Cramer-Wold device. Scheffe's theorem. Polya's theorem. Slutsky's theorem.
(2) Variance stabilizing transformations
(3) Asymptotic distribution of order statistics.
(4) Large sample properties of maximum likelihood estimates and the method of scoring.
(5) Pearson's chi-square statistic. Chi-square and likelihood ratio test statistics for various hypotheses related to contingency tables.
(6) Independence, Random walk, discrete time/discrete space Markov chains - basic theory, examples including queueing/ birth-death chains/branching processes.
  Suggested books :
(1). Approximation Theorems in Mathematical Statistics by R.J. Serfling.
(2). Linear Statistical Inference and Its Applications by C.R. Rao.
(3). Mathematical Methods of Statistics by H.Cramer.
(4). A first course in stochastic processes by S. Karlin and H.M. Taylor.

Applied Stochastic Processes (for B-stream only)
  Introduction : Brief overview of modelling -- deterministic/stochastic; discrete time / continuous time.
At least three topics from the following list, but not more than four.

(1) Branching Processes : review of discrete time branching process, extinction probabilities and asymptotic behaviour brief excursion to continuous time branching process, Two-type branching process,  Branching process with general lifetime variable (Bellman-Harris process).
(2) Modelling in Genetics : Brief review of genetics, including the Hardy-Wienberg laws, Their ramifications including mutation and fitness coefficient, Inbreeding and changes of coefficient of  inbreeding over generations,
Markovian models : sibmating,Wright-Fisher, Moran, Kimura models
W-F model with varying generation sizes, Hidden Markov models.
(3) Continuous Markov processes : Pure birth / Pure death process, Birth and death process, General theory, Multi-dimensional processes, Some non-homogeneous processes.
(4) Epidemic Modelling : Simple and General epidemics -- both deterministic as well as stochastic. Threshold theorems (with or without proof). Greenwood, Reed - Frost models, Neyman - Scott models of spatial spread of epidemics.
(5) Queueing Processes : Introduction, Markovian queueing model, Little's formula, Queues with finite capacity, Finite source queues, Tandem queues, Erlangian models, Models with general arrival and/or service patterns 
(6) Point Processes : Renewal process, Marked point process/ Compound Poisson process, Filtered point/Poisson process, Self-exciting point process, Doubly stochastic Poisson process.
Books :
  S Karlin, HM Taylor : A First Course in Stochastic Processes.
  NTJ Bailey : The Elements of Stochastic Processes.
  DR Cox, HD Miller : The Theory of Stochastic Processes.
  T E Harris : The theory of Branching Processes
  Ewens, W.J.: Mathematical population genetics
  Kingman, J.F.C. : Mathematics of genetic diversity                   
   Bailey, N.T.J : The mathematical theory of infectious diseases andits applications
  D Gross & CM Harris : Fundamentals of Queueing Theory.
  UN Bhat : Elements of Applied Stochastic Processes.
  HC Tijms : Stochastic Modelling and Analysis.
  HM Taylor & S Karlin : An Introduction to Stochastic Modelling.
  DR Cox & V Isham : Point Processes.
  DL Snyder : Random Point Processes.
  WJ Ewens and G.R. Grant : Statistical Methods in Bioinformatics.
  MS Waterman : Introduction to Computational Biology.
Optimization Techniques I
1. Review of Lagrange method of multipliers, maxima and minima of differentiable functions of several variables, some exercises
2. Constrained optimization problems, several types such LP, NLP, ILP Combinatorial Optimization Problems, several types of LP and ILP problems that occur in applications, formulation problems
3. Convex sets, flats, hyperplanes, Interior and closure, compact convex sets (The book convexity by Roger Webster, Oxford University Press, 1994 can be used)
4. Extreme points of convex sets, supporting hyperplanes, basic feasible solutions, correspondence between extreme points and basic feasible solutions (the book `Convexity by Webster, or the book non-linear programming by Bazaraa and Shetty can be used).
5. Development of the simplex method, including artificial variables in two phases (The book convexity by Webster can be used, or the book by Taha.)
6. Dual problems. A constructive proof of the duality result using the simplex tableau, Interpretation of dual variables as shadow prices on resources, complementary slackness (Taha's book can be used)
7. Branch and bound method for integer linear programming, General princilpes, Balas' implicit enumeration algorithm. (The book by Garfinkel and Nemhauser can be used)    
8. Introduction to Bellman's dynamic programming set-up, Bellman's principle of optimality, the use of this principle for solving some problems (such as the knapsack problem, shortest path problem etc.)

Statistical Inference I
(1) Game theoretic formulation of a statistical decision problem with illustration. Bayes, minimax and admissible rules. Complete and minimal complete class. Detailed analysis when the parameter space is finite.
(2) Sufficiency, minimal sufficiency and completeness. Factorization theorem. Convex loss and Rao-Blackwell theorem. Unbiased estimates and Cramer-Rao inequality. Stein estimate and shrinkage for multivariate normal mean. (If time permits: Karlin's theorem on admissibility.)
(3) Tests of hypotheses. MLR family. UMP and UMP unbiased tests. Detailed analysis in exponential models.
(4) Bayes estimates and tests. Bayesian credible region. Non-informative priors.
(5) (If time permits: Equivariance of estimates and inavariance of tests. Optimum equivariant estimates and invariant tests.)
(6) Discussion of various paradigms of statistical inference.
  Possiibe Reference Texts :
(1). Statistical Decision Theory by T.S. Ferguson
(2). Theory of Point Estimation by E.L. Lehmann
(3). Testing of Statistical Hypotehses by E.L. Lehmann
(4). Mathematical Statistics by P.J. Bickel and K.A. Doksum
(5). Statistical Decision Theory by J.O. Berger
(6). Statistical Inference by R.L. Berger G. Casella
(7). Theory of Point Estimation by E.L. Lehman and G. Casella.

Large Sample Statistical Methods
(1) Review of various modes of convergence of random variables and Central Limit Theorems. Cramer-Wold device. Scheffe's theorem. Polya's theorem. Slutsky's theorem.
(2) Asymptotic distribution of transformed statistics. Derivation of the variance stabilizing formula. Asymptotic distribution of functions of sample moments like sample correlation coefficient, coefficient of variation, measures of skewness and kurtosis, etc.
(3) Asymptotic distribution of order statistics including extreme order statistics. Bahadur's result on asymptotic behaviour of sample quantiles.
(4) Large sample properties of maximum likelihood estimates and the method of scoring.
(5) Large sample properties of parameter estimates in linear, nonlinear and generalized linear models.
(6) Pearson's chi-square statistic. Chi-square and likelihood ratio test statistics for simple hypotheses related to contingency tables. Heuristic proof for composite hypothesis with contingency tables as exampls.
(7) Asymptotic behaviour of posterior distributions and Bayes estimates, preferably without proof but using heuristic justification based on Laplace approximation.
(8) Large sample nonparametric inference (e.g., asymptotics of U-statistics and large sample distribution of various rank based statistics, large sample behaviour of Kolmogorov-Smirnov statistics).
(9) Brief introduction to locally asymptotic normal theory and asymptotic optimality.
Possible Reference Texts :
(1). Approximation Theorems in Mathematical Statistics by R.J. Serfling.
(2). Linear Statistical Inference and Its Applications by C.R. Rao.
(3). Mathematical Methods of Statistics by H.Cramer.
(4). van der Vaart.
Regression Techniques
Review of multiple linear regression, partial and multiple correlation;
Violation of linear model assumptions: consequences, diagnostics and remedy  (including properties of residuals and leverages);
Robust regression techniques : LAD and LMS regression (brief exposure);
Model building: subset selection, lack-of-fit tests;
Collinearity: diagnostics and strategies  (including ridge & shrinkage regression and dimension reduction methods);
Discordant observations: diagnostics and strategies;
Topics from:
 (i) Nonlinear and Generalized Linear Models: inference and diagnostics;
 (ii) Introduction to nonparametric regression techniques : Kernel, local polynomial and spline based methods;
 (iii) Strategies for missing and censored data;
Presentation of projects and discussion;
Data analysis with computer packages.

Suggested Text:  Modern Regression Methods (Wiley Series in Probability and Statistics, Applied Probability and Statistics), 1996, by Thomas P. Ryan; Montgomery (2nd edition), 2001, Belsley, Kuh and Welsch, Rousseau and Leroy's Robust Regression.

Time Series Analysis
Review of various components of time series, plots and descriptive statistics;
Discrete-parameter stochastic processes: strong and weak stationarity, autocovariance and autocorrelation;
Spectral analysis of weakly stationary processes: Periodogram, fast Fourier transform;
Models: Moving average, autoregressive, autoregressive moving average and autoregressive integrated moving average  processes, Box-Jenkins model, state-space model;
Linear filters, signal processing through filters;
Inference in ARMA and ARIMA models;
Forecasting: ARIMA and state-space models, Kalman filter;
Model building: Residuals and diagnostic checking, model selection;
Strategies for missing data;
Time-frequency analysis: short-term Fourier transform, wavelets;
Data analysis with computer packages.

Suggested Texts:
1. Time Series Analysis and Its Applications (Springer Texts in Statistics), 2000, by Robert H. Shumway, David S. Stoffer.
2. Introduction to Time Series and Forecasting (Springer Texts in Statistics), 2nd edition, 2002, by Peter J. Brockwell, Richard A. Davis.
3. Introduction to statistical time series (Wiley Series in Probability and Statistics), 2nd edition, 1996, by Wayne A. Fuller.

Sample Surveys and Design of Experiments
Sample Surveys (1/2 semester)

Introduction to  Unified theory of sampling, Sampling designs and sampling schemes, correspondence; Classes of estimators, Homogeneous linear estimates  and unbiasedness condition; Godambe's Non Existence Theorem, Basu's Difference Estimator; Horvitz Thompson Estimator, its variance estimators, Lahiri Midzuno Sen Scheme, nonnegativity of Sen Yates Grundye stimator: PPS Sampling, Hunsen Hurwitz estimator, Desraj's Ordered estimators for WOR selection, Murthy's Symmetrized DesRaj Estimator, Var. estimator; Brief mention of IPPS Schemes.
Double Sampling on Successive Occasions, Double Sampling for Stratification; Cost and variance functions;
Nonresponse; Hunsen Hurwitz estimator. Politz Simmons technique for Not At Home 's, RRT: Warner's model, related and unrelated questions, Nonresponse Stratum and Double sampling.
Practicals and data analytic illustrations on above topics.

Suggested Texts:
W.G. Cochran, M.N. Murthy, P. Mukhopadhyay

Design of Experiments (1/2 semester)
Review of non-orthogonal block designs under fixed effects models,connectedness,orthogonality and balance; applications; notion of mixed effects models. 
BIBD: definition, applications, analysis and efficiency, construction (only OS1 and OS2); introduction to row-column  designs and their applications.
Symmetrical factorials, fractional factorials, introduction to orthogonal arrays and their applications.
Practicals on the above topics using statistical packages for dataanalytic illustrations.
Books: Aloke Dey, Raghavarao, Dean and Voss, P W M John, M C Chakravarty

Multivariate Analysis
Review of: multivariate distributions, multivariate normal distribution and its properties, distributions of linear and quadratic forms, tests for partial and multiple correlation coefficients and regression coefficients and their associated confidence regions. Data analytic illustrations.
Wishart distribution (definition, properties), construction of tests, union-intersection and likelihood ratio principles, inference on mean vector, Hotelling's T2.
Inference on covariance matrices.
Discriminant analysis.
Basic introduction to: principal  component analysis and factor analysis.
Practicals on the above topics using statistical packages for data analytic illustrations.

Books: Anderson, Johnson and Wichert, Mardia et. al, Srivastava and Khatri

Linear Algebra and Linear Models (for NB-stream only - this course is meant to prepare students for Regression Techniques and Multivariate Analysis, to be taught in the second semester)
Linear Algebra 
Vector spaces with real and complex scalars: subspace, linear dependence and  independence, basis, dimension, sum and intersection of subspaces. Linear equations: homogeneous and nonhomogeneous systems, solution space, consistency and general solution, numerical examples. Inner product and norm: geometric interpretation, Gram-Schmidt orthogonalization process, orthogonal projection, projection on a subspace.  [As in chapters 1, 5 and 7 of LSI.]
Linear transformations and Matrices : rank, trace, elementary operations, canonical reductions, orthogonal matrices, symmetric matrices, inverse, sweep-out method. Operations with partitioned matrices. [As in Chapter 1 of LSI and Chapters 2-3 of LA.]
Determinant: definition and properties, computation. Characteristic roots and vectors: numerical examples. Quadratic forms : classification and transformations, canonical reduction of real symmetric matrices, spectral decomposition, Caley-Hamilton theorem. [As in Chapter 1 of LSI and Chapter 8 of LA.]
Generalized inverse: properties, applications. Projection operators on Rn as idempotent matrices : properties. [As in Chapter 1 of LSI and Chapter 3 of LA.]
1. Linear Statistical Inference (LSI) by C. R. Rao .
2. Linear Algebra (LA) by A.R. Rao and P. Bhimashankaram
3. R.B. Bapat

Linear Models
Linear statistical models, illustrations, Gauss-Markov model, normal equations and least squares estimators, estimable linear functions, g-inverse and solution of normal equations. Error space and estimation space. Variances and covariances of BLUEs. Estimation of error variance. Fundamental theorems of least squares and applications to tests of linear hypotheses. [As in Chapter 4 of LSI and Chapters 1 and 2 of CLM.]
Fisher-Cochran theorem, distribution of quadratic forms. [As in Chapter 3 of LSI.]

1. LSI by C. R. Rao,
2. A course in Linear models by A.M. Kshirsagar (CLM),
3. Linear estimation and Design of Experiments by D.D. Joshi.

Discrete Mathematics
(a) Generating functions, recurrence relations, Polya's theory of counting, Ramsey theory.
(b) Graphs, connectedness, paths, cycles. Regular graphs. Strongly regular graphs. Eigenvalues of graphs. Peron-Frobenius theorem. Characterization of connectedness byspectrum. Classification of graphs with largest eigenvalue atmost 2 (Dynkin diagrams).
(c) Steiner 2-designs and their strongly regular graphs. Steiner t-designs. Witt designs and Golay codes.
(d) Probabilistic Methods in Combinatorics and Number Theory, Second Moment method, FKG inequalities, superconcentrators, random graphs (if time permits).

1. Designs, Graphs, codes and their links by P.J. Cameron and J.H. VanLint, London Math.Society, Students text #22, Cambridge University press,1991.
2. The Probabilistic Method by Alon & Spencer, Wiley Interscience.
3. Random Graphs by Balabas, Academic Press.
4. Random Algorithms by Motwani and Rahgavan, Cambridge University Press.

Nonparametric and Sequential Analysis (for NB-stream only)
Nonparametric Methods: Formulation of the problems, order statistics and their distributions. Tests and confidence intervals for population quantiles. Sign test. Test for symmetry, signed rank test, Wilcoxon-Mann-Whitney test, Kruskal-Wallis test. Run test, tests for independence. Concepts of asymptotic efficiency. Estimation of location and scale parameters.
Sequential Analysis: Need for sequential tests. Wald's SPRT, ASN, OC function. Stein's two stage fixed length confidence interval. Illustrations with Binomial and Normal distributions. Elements of sequential estimation.
Practicals using statistical packages.

Reference texts:
1.Nonparametrics: Statistical Methods Based on Ranks by E.L. Lehmann.
2.Theory of Rank Tests by J. Hajek and Z. Sidak.                       
3.Mathematical Statistics by P.J. Bickel and K.A. Doksum.
4.Statistical Inference by G. Casella and R.L. Berger.

Programming and Data Structures (for NB-stream only)
Programming in a structured language such as C.
Data Structures: definitions, operations, implementations and applications of basic data structures. Array, stack, queue, dequeue, priority queue, doubly linked list, orthogonal list, binary tree and traversal algorithm, threaded binary tree, generalized list.
Binary search, Fibonacci search, binary search tree, height balance tree, heap, B-tree, digital search tree, hashing techniques.
Reference Texts:
1. The C Programming Language by Brian W. Kernighan and Dennis M. Ritchie.
2. Theory and Problems of Programming with C by Byron S. Gottfried (Schuam's Outline Series).
3. Data Structures and Algorithms by A. Aho, J. Hopcroft and J. Ullman.
4. Data Structure Techniques by T.A. Standish.
5. Data Structures using PASCAL by A.M. Tanenbaum and M.J. Augesestein.

Second Year Courses

AP Courses

Advanced Probability I
(a) Radon Nikodym Theorem. Conditional Expectation. Regular conditional probability. Relevant measure theoretic development.
(b) Finite and infinite products. Kolmogorov-Tulcea Theorems.
(c) Discrete parameter martingales with various applications including U-statistics. Path properties of continuous parameter martingales.

Functional Analysis (see MSP syllabi)

Stochastic Processes I
Selected topics from the following :
(a) Weak convergence of probability measures on polish spaces including C[0,1].
(b) Brownian motion : Construction, simple properties of paths. Brief introduction to Stochastic Calculus.
(c) Markov processes and generators.

Stochastic Processes II
Poisson Process, point processes, infinite particle systems, interacting particle systems.

Ordinary and Partial Differential Equations
Linear ODE, Power series method and orthogonal polynomials, Picard's theorem, generalities of PDE; Heat, Laplace and Wave equations, Initial value problems and boundary value problems.

Advanced Functional Analysis
Any of the following topics or some other general area of functional analysis: (a) Geometry of Banach Spaces. Choquet's theory ofintegral representation of compact convex sets, basis in Banach spaces, Weak Compactness, Vector measures. (b) Topics in Operator Theory. Direct integral decomposition of unitary operators, operator models (Nagy-Poias), functional calculus for commuting operators.

Ergodic Theory
(a) Measure-preserving transformations, recurrence, ergodicity, ergodic theorems, mixing. (b) Isomorphism, conjugacy and spectral isomorphism. (c) Measure-preserving transformations with discrete spectrum. Eigen values and eigen functions. Discrete spectrum. Group rotations and Halmos-Von Neumann representation theorem. (d) Entropy : entropy of a partition, entropy of a measure-preserving transformation, methods of calculating entropy, Kolmogorov-Sinai theorem, Bernoulli automorphisms, Kolmogorov automorphism.

Quantum Probability
Events, observables and states - Gleason's theorem. Expectation, variance and moments. Heisenberg's uncertainty principle. Evolution as a one-parameter unitary group, Schrodinger and Heisenberg equations. Tensorproducts, symmetric and antisymmetric tensor products, Boson and Fermion Fock spaces. Weyl representation, geometric derivation of infinitely divisible distributions and processes with independent increments. Creation, conservation and annihilation processes, quantum stochastic integrals and quantum Ito's formula. Solution of quantum stochastic differential equations with bounded constant operator coefficients. (Prerequisite : Operator theory in Advanced Functional Analysis).

AS Courses

Actuarial Methods 
1. M-I course Statistical Inference I,
2. M-I course Time Series Analysis,
3. M-I course Regression Techniques.
(a) Review of Decision theory and actuarial applications.
(b) Loss distributions: modelling of individual and aggregate losses, moments, fitting distributions to claims data, deductibles and retention limits, proportional and excess-of-loss reinsurance, share of claim amounts, parametric estimation with incomplete information.
(c) Risk models: models for claim number and claim amount in short-term contracts, moments, compound distributions, moments of insurer’s and reinsurer’s share of aggregate claims.
(d) Review of Bayesian statistics/estimation and application to credibility theory.
(e) Experience rating: Rating methods in insurance and banking, claim probability calculation, stationary distribution of proportion of policyholders in various levels of discount.
(f) Delay/run-off triangle: develoment factor, basic and inflation-adjusted chain-ladder method, alternative methods, average cost per claim and Bornhuetter-Ferguson methods for outstanding claim amounts, statistical models.
(g) Review of generalized linear model, residuals and diagnostics, goodness-of-fit, applications.
(h) Review of time series analysis, filters, random walks, multivariate AR model, cointegrated time series, non-stationary/non-linear models, application to investment variables, forecasts.
(i) Assessment of methods through Monte-Carlo simulations.
`Actuarial Mathematics' by N.L. Bowers, H.U. Gerber, J.C. Hickman, D.A. Jones, and C.J. Nesbitt, (2nd ed), Society of Actuaries, 1997.
`Loss Models: From Data to Decisions' by S.A. Klugman, H.H. Panjer, G.E. Willmotand G.G. Venter, John Wiley & Sons, 1998.
`Practical Risk Theory for Actuaries' by C.D. Daykin, T. Pentikainen and M. Pesonen, Chapman & Hall, 1994.

Life Contingencies 
(a) Assurance and annuity contracts: definitions of benefits and premiums, various types of assurances and annuities, present value, formulae for mean and variance of various continuous and discrete payments, various conditional probabilities from ultimate and select life tables, mthly payments, related actuarial symbols, inter-relations of various types of payments.
(b) Calculation of various probabilities from life tables: notations, probability expressions, approximations, select and ultimate tables, alternatives to life tables.
(c) Calculation of various payments from life tables: principle of equivalence, net premiums, prospective and retrospective provisions/reserves, recursive relations, Thiele’s equation, actual and expected death strain, mortality profit/loss.
(d) Adjustment of net premium/net premium provisions for increasing/decreasing benefits and annuities: actuarial notations, calculations with ultimate or select mortality, with-profits contract and allied bonus, net premium, net premium provision.
(e) Gross premiums: Various expenses, role of inflation, calculation of gross premium with future loss and equivalence principle for various types of contracts, alternative principles, calculation of gross premium provisions, gross premium retrospective provisions, recursive relations.
(f) Functions of two lives: cash-flows contingent on death/survival of either or both of two lives, functions dependent on a fixed term and on age.
(g) Cash-flow models for competing risks: Markov model, dependent probability calculations from Kolmogorov equations, transition intensities.
(h) Use of discounted emerging costs in pricing, reserving and assessing profitability: unit-linked contract, expected cash –flows for various assurances and annuities, profit tests and profit vector, profit signature, net present value and profit margin, use of profit test in product pricing and determining provisions, multiple decrement tables, cash-flows contingent on multiple decrement, alternatives to multiple decrement tables, cash-flows contingent on non-human life risks.
(i) Cost of guarantees: types of guarantees and options for long term insurance contracts, calculation through option-pricing and stochastic simulation.
(j) Heterogeneity in mortality: contributing factors, main forms of selection, selection in insurance contracts and pension schemes, selective effects of decrements, risk classification in insurance, role of genetic information, single figure index, crude index, direct/indirect standardization, standardized mortality/morbidity ratio (SMR).
`Actuarial Mathematics' by N.L. Bowers, H.U. Gerber, J.C. Hickman, D.A. Jones, and C.J. Nesbitt, (2nd ed), Society of Actuaries, 1997.
`Life Contingencies' by A. Neill, Heinemann, 1977.
`The Analysis of Mortality and Other Actuarial Statistics' by B. Benjamin and J.H. Pollard, 3rd ed., Institute of Actuaries and Faculty of Actuaries, 1993.
`Modern Actuarial Theory and Practice' by P.M. Booth, R.G. Chadburn, D.R. Cooper, S. Haberman and D.E. James, Chapman & Hall, 1999.

Actuarial Models 
Prerequisite:  B-III course Introduction to Stochastic Processes (for B-stream)
OR M-I course Large Sample Theory and Markov Chain (for NB-stream).
Corequisite:  M-II course Survival Analysis.
(a) 1. Principles of actuarial modelling: model selection, interpolation/extrapolation issues and diagnostics.
(b) Review of various types of stochastic processes; their actuarial applications.
(c) Review of Markov chain; frequency based experience rating and other applications.
(d) Markov process (Poisson process, Kolmogorov equations, Illness-Death and other survival models, effect of duration of stay, Markov jump processes).
(e) Review of survival models, future life random variable and related actuarial notations, two-state model for single decrement.
(f) Review of nonparametric estimation and Cox model-based regression.
(g) Models of transfer between multiple states: General Markov models of transfers, standard actuarial notations for transfer probabilities and rates, their equations.
(h) Estimation of transition intensities: MLE under piecewise constant assumption, Poisson approximation.
(i) Central Exposed to Risk: data type, computation, estimation of transition probabilities, census approximation of waiting times, rate intervals, census formulae for various definitions of age.
(j) Graduated estimates: reasons for comparison of crude estimates of transition intensities/probabilities to standard tables, statistical tests and their interpretations, test for smoothness of graduated estimates, graduation through parametric formulae, standard tables and graphical process, modification of tests for comparing crude and graduated estimates and to allow for duplicate policies.
`Actuarial Mathematics' by N.L. Bowers, H.U. Gerber, J.C. Hickman, D.A. Jones, and C.J. Nesbitt, (2nd ed), Society of Actuaries, 1997.
`Modeling, Analysis, Design, and Control of Stochastic Systems' by V.G. Kulkarni, Springer, 1999.
`Probability and Random Processes' by G. Grimmett and D. Stirzaker, (3rd ed), Oxford University Press, Oxford, 2001.
`Analysing Survival Data from Clinical Trials and Observational Studies' by E. Marubini and M.G. Valsecchi, John Wiley & Sons, 1995.

Survival Analysis
(a) Introduction: Survival data, hazard function (continuous and discrete),
(b) Nonparametric inference: Kaplan-Meier estimate, Nelson-Aalen estimate.
(c) Comparison of survival curves.
(d) Survival models: Exponential, Weibull, log-normal, gamma etc. regression models, Proportional hazards model.
(e) Parametric inference: Censoring mechanisms and likelihood, large-sample likelihood theory, iterative methods for solution,
(f) Binomial and Poisson models for discrete data.
(g) Proportional Hazard model: Methods of estimation, estimation of survival functions, time-dependent covariates.
(h) Markov Models: Two-state model, illness-death model, maximum likelihood estimator and its properties.
(i) Rank tests with censored data.
(j) Survival data with competing risks.

Statistical Computing (see ASDA syllabi)

Game Theory I (see QE syllabi)

Microeconomic Theory I (see QE syllabi)

Macroeconomic Theory I (see QE syllabi)

Theory of Finance I
Prerequisite:  M-II course Game Theory I.
(a) Choice under uncertainty and stochastic dominance, Mean – Variance portfolio theory leading to the Capital Asset Pricing Model, two fund separation and linear valuation, Multifactor models, elements of arbitrage pricing theory.
(b) Elements of stochastic processes, second order processes, continuity, integration, differentiation, stochastic differential equations of the first and second order.
(c) Derivatives, hedging strategies, Greeks, option pricing, risk neutral pricing, forwards and futures, term structure of interest rates, swaps, Binomial trees, Black-Scholes analysis, alternatives to Black-Scholes, management of market risk (VaR etc.).
(d) Structure of stock markets in general and USA in particular, definition and testing of different levels of efficient market hypothesis, regulations, role of different agents.
(a)`Foundations of Mathematical Finance' by Huang and Litzenberger; `Theory of Financial Decision Making' by J.E. Ingersoll.
(b)`Introduction to Stochastic Processes' by Hoel, Port and Stone; `Introduction to Mathematics of Finance, I" by Karatzas; `Stochastic Differential Equation' by B. Oksendal.
(c) `Options, Futures and other Derivatives' by J.C. Hull; `Discrete Time Mathematical Finance' by A. Pliska; `Derivative Securities' by Jarrow and Turnbull.
(d) `How the Stock Market Works' by J. Dalton.
Several Papers (Copies to be supplied).

Theory of Finance II
Prerequisite:  M-II course Game Theory I.
(a) Corporate finance, mainly valuation of assets, time value, selection of projects, debt-equity choice, pecking order hypothesis, budgeting, corporate structure, tax regulations and governance, agency problems, separation of ownership and control. Stock market operation including Initial Public Offering (IPO).
(b) Banking finance including regulations, structure of banks, market imperfections and need for financial intermediaries, lender-borrower relationship.
(c) Indian financial System, banking sector, NBFCs, RBI and SEBI, securities and money market structure, regulations, development of stock markets.
(a) `Financial Market analysis' by D. Blake; `Principles of Corporate Finance' by Brealey and Myers; Several Papers (Copies to be supplied).
(b) `Microeconomics of Banking' by Freixas and Rochet.
(c) `Indian Financial System' by H.R. Machiraju.

Applied Multivariate Analysis (see ASDA syllabi)

Life Testing and Reliability (see ISOR syllabi)

Theory of Games and Statistical Decisions (see MSP syllabi)

Stochastic Processes I (see AP syllabi)

Econometric Methods (see QE syllabi)

Statistical Methods in Demography (see BSDA syllabi)

ASDA Courses

Advanced Design of Experiments (see ISOR syllabi)

Analysis of Discrete Data
Measures of association.
Structural models for discrete data in two or more dimensions.
Estimation in complete tables. Goodness of fit, choice of a model.
Generalized Linear Model for discrete data, Poisson and Logistic regression models.
Log-linear models.
Elements of inference for cross-classification tables.
Models for nominal and ordinal response.
Data Analysis with computer packages.
Books: Agresti, Tibshirani et al.

Statistical Computing
(a) Review of simulation techniques for various probability models, and resampling
(b) Computational problems and techniques for

(c)Analysis of incomplete data: EM algorithm, single and multiple imputation
(d) Markov Chain Monte Carlo and annealing techniques
(e) Neural Networks, Association Rules and learning algorithms

Suggested texts:
1. "Simulation" by S.M. Ross (Second edition, Wiley, 1997).
2. "Elements of Statistical Computing" by R.A. Thisted (Chapman and Hall, 1988).
3. "Modern Applied Statistics with S-Plus" by W.N. Venables and B.D. Ripley Third Edition, Springer, 1999).
Rousseau-Leroy, McCullagh-Nelder, CART, Brian Everitt, Rubin, Little-Rubin
4. "The Elements of Statistical Learning: Data Mining, Inference and Prediction"
by Hastie, Tibshirani and Friedman (Springer, 2001).

Advanced Sample Surveys

(a) Unified theory of sampling, non-existence theorems relating to labelled populations. Traditional model-based and Bayesian theories of inference in finite population sampling. Sufficiency, Bayesian sufficiency, completeness. Optimal and various other useful sampling strategies. Integration of different principles and methods of sampling in adopting composite sampling procedures in actual practice.
(b) Randomized response technique, post-stratification, small area estimation, synthetic estimation, repeated sampling, balanced replication, Jack-Knifing.
(c) Organizational aspects of planning large-scale sample surveys, non-sampling errors, non-response. Familiarity with NSS work and some specific large-scale surveys. 

Applied Multivariate Analysis
Graphical representation of multivariate data.
Dimension reduction methods : Review of Principal Component and Factor Analyses, Canonical Correlation analysis, Correspondence Analysis, Multidimensional Scaling.
Classification methods: Review of Discriminant Analysis, Cluster analysis.
Nonparametric and robust methods of  multivariate analysis.
Data analysis with relevant statistical packages.

Survival Analysis (see AS syllabi)

Statistical Methods in Genetics I (see BSDA syllabi)

Discrete and continuous time stochastic models, diffusion equation, stochastic models for population growth and extinction (includes branching process), interacting population of species - competition and predation, chemical kinetics, photosynthesis and neuron behaviour. Deterministic and stochastic modelsfor epidemics and endemics, interference models, vaccination models, geographical spread, parasitic diseases, parameter estimation related to latent, infection and incubation periods. Bioassay. Case studies.

Life Testing and Reliability (see ISOR syllabi)

Theory of Games and Statistical Decisions (see MSP syllabi)

Econometric Methods (see QE syllabi)

Quantitative Models in Social Sciences
Selected topics from the following and/or any other areas in social sciences of statistical relevance.
(a) Psychology : Stochastic models for learning. Models for choice behaviour. Stochastic models for test scores.
(b) Sociology : Latent structure models. Applications of graph theory; preference, social interactions, indifference, energy modelling, social inequalities etc. Structural models.
(c) Economics : Demand models, income inequality, etc.
(d) Management science : Inventory models. Scheduling. Queues.

Pattern Recognition and Image Processing
Pattern Recognition
Review of Bayes classification: error probability, error bounds, Bhattacharya  bounds, error rates and their estimation;
Parametric and nonparametric learning, density estimation;
Classification trees;
k-NN rule and its error rate;
Neural network models for pattern recognition: learning, supervised and  unsupervised classification;
Unsupervised classification: split/merge techniques,  hierarchical clustering algorithms, cluster validity, estimation of mixture distributions;
Feature selection: optimal and suboptimal algorithms;
Some of the other approaches like the syntactic,  the fuzzy set theoretic,  the neurofuzzy,  the evolutionary (that is, based on genetic algorithms), and applications; Some recent topics like data mining, support vector machines, etc.
Introduction to Statistical Pattern Recognition, 2nd edition, by K. Fukunaga, New York: Academic Press, 1990. 
Pattern Recognition: A Statistical Approach by P A Devijver and J Kittler, London: Prentice-Hall, 1982. 
Algorithms for Clustering Data, by A K Jain and R C Dubes, Englewood Cliffs, NJ:  Prentice-Hall, 1988. 
Cluster Analysis, by B S Everitt, New York: Halsted Press, 1993. 
Syntactic Pattern Recognition and Applications, by K S Fu,  Englewood Cliffs, NJ: Prentice-Hall, 1982. 
Pattern Recognition with Fuzzy Objective Function  Algorithms, J.C.Bezdek, New York: Plenum Press, 1981. 
Elements of Statistical Learning, by T. Hastie, R. Tibshirani and J. H. Friedman, New York: Springer-Verlag, 2001. 
Pattern Recognition and Neural Networks by B D Ripley, New York: Cambridge University Press, 1996. 
Pattern Recognition by S Theodoridis and K Koutroumbas, San Diego: Academic Press, 1999. 

Image Processing

Introduction, image definition and its representation;
Typical IP operations like enhancement, contrast stretching,  smoothing and sharpening, greylevel  thresholding,  edge detection,  medial axis transform, skeletonization/ thinning, warping;
Segmentation and pixel classification;
Object recognition;
Some statistical (including Bayesian) approaches for the above, like Besag's ICM algorithm,  deformable templates approach of Grenander and colleagues, and so on.
Handbook of Pattern Recognition and Image Processing, vols. 1 & 2, by T Y Young and K S Fu,  New York: Academic Press, 1986.
Fundamentals of Digital  Image Processing, by A Jain, New Delhi: Prentice-Hall, 1989. 
Digital Image Processing, by K R Castleman, Englewood Cliffs, NJ: Prentice-Hall, 1996. 
Statistics and Images, by K V Mardia and G K Kanji (eds.), Abingdon: Carfax, 1993.

BSDA Courses

Statistical Methods in Genetics I
Mendel's laws, Estimation of allele frequencies, Hardy-Weinberg law, Mating tables, Snyder's ratios, Models of natural selection and mutation, Detection and estimation of linkage (recombination), Inheritance of quantitative traits, Stochastic models of carcinogenesis.

Analysis of Discrete Data (see ASDA syllabi)

Statistical Computing (see ASDA syllabi)

Survival Analysis (see AS syllabi)

Statistical Methods in Public Health
Longitudinal data analysis (Repeated measures design, Growth models, Regression models, etc.).
Epidemiology (Case-control studies, Estimation of prevalence and incidence, Age at onset distributions, Assessing spatial and temporal patterns, etc.).
Theory of epidemics (Simple and general epidemics, Recurrent epidemics and endemicity, Discrete-time models, Spatial models, Carrier models, Host - vector and venereal disease models, etc.).

Statistical Methods in Biomedical Research  
Bioassay ( Direct and indirect assays, Quantal and quantitative assays, Parallel line and slope ratio assays, Design of bioassays, etc.)
Clinical trials (Different phases, Comparative and controlled trials, Random allocation, Parallel group designs, Crossover designs, Symmetric designs, Adaptive designs, Group sequential designs, Zelen's designs, Selection of subjects, Ethical issues, Outcome measures, Protocols, Sample size determination, etc.).
Review of theory and application of Generalized Linear Model Quasilikelihood and Generalized Estimating Equations, Correlated binary data, overdispersion.

Advanced Design of Experiments (see ISOR syllabi)

Advanced Sample Surveys (see ASDA syllabi)

Applied Multivariate Analysis (see ASDA syllabi)

Life Testing and Reliability (see ISOR syllabi)

Theory of Games and Statistical Decisions (see MSP syllabi)

Statistical Ecology
Population dynamics, Spatial patterns in one-species populations, Spatial relations of two or more species, Many-species populations.

Statistical Methods in Genetics II
1. Evolution of DNA sequences - Kimura's two-parameter and Jukes-Cantor models.
2. DNA sequence alignment - Needleman-Wunsch and Smith-Waterman algorithms.
3. basisc Local Alignment Search Tool.
4. Gene trees and species trees.
5. Estimation of evolutionary and population genetic parameters from DNA sequence data.
6. Gene mapping methodologies: (a) transmission-disequilibrium test, (b) linkage disequilibrium mapping, (c) quantitative trait locus mapping.
7. Genome databases.

Statistical Methods in Demography

Sources of demographic data, Mortality analysis and models, Population growth, composition and distribution, Projection models, Stable population theory, Life tables, Health statistics.

Pattern Recognition and Image Processing (see ASDA syllabi)

Special Topics in BSDA
Topics may be chosen from the following list:
Risk Assessment, Analysis of Family Data, Mathematical Models, in Biology, Multi-point mapping, Environmental Statistics, Statistical Methods in AIDS Research, Statistical Applications in Molecular Biology.

ISOR Courses

Advanced Design of Experiments
(a) Optimality criteria, A-, D-, E-optimality, universal optimality of BBD and generalized Youden Square Designs.
(b) Orthogonal arrays, Rao's bound, construction, Hadamard matrices.
(c) Orthogonal arrays as fractional factorial plans, main effect plans for 2-level factorials.
(d) Response surface designs, method of steepest ascent, canonical analysis and ridge analysis of fitted surface.
(e) Robust designs and Taguchi methods.
Topics from the following:
(f) Mixture experiments.
(g) Asymmetric factorials, orthogonal factorial structure, Kronecker calculus for factorials, construction.
(h) Optimal regression designs for multiple linear regression and quadratic regression with one explanatory variable, introduction to D-optimal design measure.
(i) Cross-over designs, applications, analysis and optimality.
(j) PBIB designs with emphasis on group divisible designs.
(k) Nested designs.

Life Testing and Reliability
(a) Different life-time distributions and their properties, Mean time between failures, hazard rates, different failure  models and the test for their validity, problems of inference (estimation and testing) for the parameters of common life-time distributions, estimation from uncensored and censored samples, concepts of accelerated life-testing, Bayesian approach to reliability estimation.
(b) Fault tree analysis, coherent systems, basic concepts of component and system reliability. Reliability bounds. Nonparametric classes of life-time distributions. Maintained systems and system availability. Special topics (models for software reliability, multivariate life distributions, systems with redundancy, stress-strength models etc).

Quality Control and Its Management
1. Control Charts: Statistical Process control and different types of control charts (including control chart for short run process, group control chart, EWMA chart, modified control chart, R&R study, Bayesian  Control charts Process capability analysis, capability indices).
2. Acceptance sampling: : Single, double, multiple, sequential and published plans; continuous, chain and bulk sampling plans, Bayesian sampling plans
3. Management of Quality Control : Concepts of quality planning, control, assurance and improvement, Elements of quality management system, TQC, TQM  quality circle, system standards, organization of six-sigma programmes in the industry.  Role of statistical techniques in quality management.

Management Applications of Optimization (should follow in the fourth semester, after Optimization Techniques II).
1. Application of linear programming to transportation problems. The properties of basis matrices arising in the transportation problem and their implications to the pivot algorithms. The stepping stone and u-v methods of solving a transportation problem.
2. The use of labeling algorithm for constructing a maximal independent set of admissible cells and its use in solving the bottle-neck assignment problem. The linear assignment problem and its solution by the Hungarian Method. The primal-dual approach to solving a transportation problem.
3. Out of kilter method for maximum commodity flow through a given network at minimal cost.
4. CPM and PERT methods for analyzing a project and deriving the project cost curve.
5. Branch and bound method (revision) and cutting plane methods for solving an integer- programming problem. Various applications of integer programming and combinatorial optimization.  The traveling salesman problem and its applications. Use of heuristic  algorithms.
6. Bellman's optimality principle and its use in solving optimization problems.
7. Portfolio selection problem and the use of Lemke's method for portfolio selection and the mean-variance model. Stochastic programming models for financial risk management.

Industrial Applications of Stochastic Processes
1. Queueing Theory:  Introduction to queuing problems. Revision of Markov chains and processes. (Also discuss the examples of queues and an inventory problem presented in Karlin and Taylor's book on applied stochastic processes).Markov Chain models for simple queueing processes. The M/M/C model, M/M/C model with finite waiting space, and models with a finite source of customers,(machine interference problem). Little's formula, Markovian Queues in tandem, Semi-Markovian queueing systems: M/G/1, M/G/1 with service vacationsandG/M/1. Batch arrivals and bulk queues. The notion of queueing network. Jackson and Kelly networks. The use of queueing network in manufacturing systems.(Sections from the book by Sheldon and Ross and the book by Kulkarni may be used).
2. Inventory control: Elementary models. The newspaper vendor problem and related models. The (s,S) policies and periodic review models. Models for perishable items. Concepts of supply-chain management.
3. Replacement: Replacement of items that fail gradually and stochastically. The use of renewal theory in comparing replacement policies.
4. Blackwell model of dynamic programming. Discounted and limiting average payoff. Applications.
5. Time series models and their use in forecasting. (depends on interest and if the optional subject on time series analysis has not been taken by the students).

Optimization Techniques II
1. Convex sets and their elementary topological properties, extreme points, simplex method (quick revision).
2. Separation theorems for convex sets and theorems of the alternative.
3. The derivation of the duality theorems of linear programming using theorems of the alternative. The dual simplex and primal dual algorithms. Sensitivity and post optimal analysis.
4. Degenaracy and cycling. Anticycling pivot selection rules in the simplex type methods.
5. Issues of computational complexity and the need for alternative methods. Klee_Minty example. Interior point methods for linear programming (Karmarkar's). (The recent edition of Taha's book has a well-illustrated section on Karmarkar's algorithm)
6. The problem of finding a maximum flow through a given network using labeling algorithm. The max-flow-min-cut theorem. Generalizations to networks with multiple sources and sinks. Feasibility theorem for circulation and its application. The Konig-Egervary theorem.
7. Optimization involving differentiable functions. Karush-Kuhn-Tucker necessary optimality conditions under constraint qualifications.  Convex and generalized convex functions and the sufficiency of KKT conditions.
8.  The linear complementarity problem as a unifying format. Lemke's algorithm.

Optimization Techniques III
Formulating Models in integer and binary variables. Integer programming and graphs. Matching and covering on graphs. Enumeration Methods. Cutting plane methods. Knapsack and group Knapsack problem. Integer programming over cones. Set covering and partitioning problem. Travelling salesman problem. Dynamic programming for various discrete optimization problem.

Optimization Techniques IV
Caratheodory, Krein-Millman, and other separation theorems for convex sets. Theorems of alternatives. Convex functions. Properties of differentiable convex sets. Programming problem with a differentiable objective function and differentiable constraint functions Kuhn-Tucker necessary conditions. Fritz-John necessary conditions. Some constraint qualifications and their geometry. A problem with a differentiable convex objective function {\it LCP} formulation of quadratic programming problem. Lenke-Howson algorithm. Kanash-Kuhn-Tucker conditions for a programming problem with a pseudo-convex objective function and differentiable quasi-convex constraint functions. Fractional {\it LP} and Its {\it LCP} formulation. Sequential unconstraint minimization technique.

Network Analysis
Definition and formulations. Maximum value flow algorithm for single commodity pure network flow problem, primal dual algorithm for transportation and assignment problem shortest chain algorithms, minimum spanning tree algorithms, single commodity minimum cost flow algorithm, critical path method, generalized network flow problems, multi-commodity flow problems, dynamic network flow algorithm and routing applications.

Sampling Inspection
Operating characteristic function Hamaker's conjecture, curtailed inspection. Switching rules. Dodge and Romig's systems and its generalizations. Bayesian sampling plans. Minimax regret and other sampling plans.

Scheduling Theory
Deterministic scheduling problems, single machine problems, branch and bound approach. Sequence-dependent set-up times and `Travelling Salesman problem. Parallel machines problem. Flow shop scheduling and job shop scheduling.

Industrial Engineering and Management

Operations management. Management time study, work sampling, incentive and job evaluation. Materials management value analysis, plan layout and inventory management. Organization staff line relationship, communication, administration, management of change. Cost of budgetary control-cost accounting, control of materials, labour and overhead, incentive schemes. Market research.

MSP Courses

Advanced Probability I (see AP syllabi)

Functional Analysis
(a) Basic metric spaces and locally compact Hausdorff spaces. Riesz representation theorem and Stone-Weierstrass theorem.
(b) Three fundamental principles of Banach Spaces (Hahn-Banach, Uniform boundedness and open mapping theorems).
(c) Hilbert spaces, operators. Spectral theorem.

Stochastic Processes I (see AP syllabi)

Statistical Inference II
Overview of classical inference. Selected topics from the following:
(a) Principles of data reduction (i) Sufficiency : Results of Halmos-Savage, Basu-Ghosh, and Bahadur. Brief discussion on undominated cases. Applications. (ii) Invariance : Invariant decision rules, equivariant estimation, invariant tests; discussion on admissibility, minimax property etc. of invariant rules.
(b) Foundations of statistics : Coherence, Bayesian analysis, Likelihood principle (results and concepts of Barnard, Birnbaum, Basu, etc.).
(c) Advanced and current topics in the frequentist and Bayesian theory of estimation and tests of statistical hypotheses.

Nonparametric Inference
Selected topics from the following : (a) Estimation : U-statistics and the corresponding asymptotic theory, Von-Mises' functional, Jack-Knife and Boot-strap methods, asymptotic efficiency. (b) Rank tests, permutation tests, asymptotic theory of rank tests under null and alternative (contiguous) hypotheses, asymptotic efficiency. (c) Nonparametric regression. (d) Bayesian nonparametric analysis. (e) Estimation of density function. (f) Robustness, M-estimates. (g) L-estimates.

Advanced Design of Experiments (see ISOR syllabi)

Advanced Sample Surveys (see ASDA syllabi)

Theory of Games and Statistical Decisions
(a) Game theory as a tool for making statistical decisions; Elements of theory of two person zero-sum games and minimax theorem; (Ref. Blackwell and Girshick; Ferguson).
(b) Theory of Statistical Decisions (detailed discussion for the general parameter and the action spaces) : (i) Randomization, Optimality, Bayes rules, minimax rules, admissible rules. Invariance and sufficiency. Complete class and essential complete class of rules. Examples. Topology of the set of randomized decision rules. (ii) Minimax Theorems. (iii) Complete class theorem. (iv) Results on admissibility and minimaxity.
Ref . Blackwell and Girshick, Ferguson, Wald, Brown, Berger.
`Loss models: from Data to Decisions' by Stuart Klugman et al. (Chapman and Hall, 1994).

Sequential Analysis and Optimal Stopping
Selected topics from the following : (a) SPRT; its optimality, OC and ASN.(b) Invariant SPRT and their termination probabilities. Sufficiency and invariance in sequential analysis. Stopping time of invariant SPRT. (c) Non-linear renewal theorem and its application in sequential analysis. (d) Stopping time, principle of backward induction, monotone case, extended stopping time and triple limit theorem. (e) Kiefer-Weiss problem, use of heat equation in stopping problems, bandit problems.

Topics in Bayesian Inference
Selected topics from the work of de Finneti, Savage, Lindley, Basu, Berger and others.

Time Series Analysis (see First Year syllabi)

Asymptotic Theory of Inference
Selected topics from the following : (a) Parametric and semi-parametric estimation (Ref. Ibragimov and Khasminski); contiguity and related results. (b) Asymptotic most powerful tests (parametric). (c) Consistency of maximum likelihood estimates (Bahadur-Wald). (d) Large deviations and Bahadur efficiency. (e) Asymptotic Theory of nonparametric estimation (if the course on nonparametric inference is not offered). (f) Berry-Esseen bound and related results. (g) Martingale approach to inference. (h) Inference for Stochastic Processes (Branching processes, diffusion processes, queuing models, etc.).

Pattern Recognition and Image Processing (see ASDA syllabi)

Statistical Computing (see ASDA syllabi)

Topology and Set Theory
Topological spaces, continuity, countability axioms.
Subspaces, products, quotients. Weak topology generated by a family of maps.
Separation properties including Urysohn and Tietze extension theorem, etc.
Compactness, Tychonoff's theorem; one-point and stone-Cech compactification.
Connectedness, path-connectedness
Metrization theorem of  Urysohn. Completeness.
Nets and filters convergence in compact spaces
Definition and examples of manifolds.

Graph Theory and Combinatorics
Elementary principles of combinatorics; inclusion and exclusion. Ramsey problem.
Graphs : isomorphism, adjacency and incidence matrices, degree sequence, Havel-Hakimi Theorem. Erdos-Gallai Theorem.
Paths, connectedness, trees; characterizations. Minimal connector problem. Kruskal's algorithm. Cut vertices, cut edges. Examples. Euler chain characterization theorem; examples. Hamiltonian graph. Different degree sequences vs. Hamiltonicity. Travelling salesman problem. Applications.
Edge colouring, vertex colouring numbers. Total chromatic number. Vizing-Gupta theorem. Bipartite graphs. Applications.
Planar graphs. Subdivision, Kuratowski's characterization. Examples. The five-colour theorem and the statement of the four-colour theorem.
Directed graphs. Applications.
Flows, cuts, max-flow min-cut theorem; applications and examples.

Advanced Algebra
Selected topics from the following :
(a) Theory of modules. Left and right modules, submodules and quotient modules, module homomorphisms, direct summand product, free modules, exact sequences. Tensor products over commutativerings, groups of homomorphisms and their principal properties.
(b) Commutative Algebra : Ideals, prime ideal, maximal ideal, nilradical and Jacobian radical, operations on ideals, extension and contraction. Rings and modules of fractions, local properties, extended and contracted ideals inrings of fractions. Primarydecomposition. Chain conditions, Noetherian rings,Certin rings, Discrete valuation rings and Dedekind domains.
(c) Galois Theory : Field extensions,fundamental theorems, splitting fields. The Galois group of a polynomial, finite fields, separability, cyclic extensions, cyclotomic extensions. solvableand radical extensions, Kummer theory. 3 (d) Representations of finite groups:
Concept of representation. Complete reducibility, uniqueness of decomposition. Group ring and regular representation, space of class functions, orthogonal relations, induced characters, induced representations, positive decomposition of regular character, Brauer's theorem.

Harmonic Analysis
(a) Fourier Series : Definitions and simple consequences. Convolutions, summability kernels. Summability (Cesaro and Abel) in norm and pointwise. Lebesgue's differentiation theorem. Order of magnitude of Fourier coefficients. Fourierseries of square summable functions. Absolutely convergent Fourier Series. Fourier-Stieltjes coefficients, positive-definite sequences, Helglotz theorem.
(b) Convergence of Fourier Series in norm, pointwise. Hardy's theorem.
(c)Fourier transform on the line, convolutions, inversion theorem, Plancherel theorem, Schwartz space and Tempered Distributions.
(d) Conjugate function and functions analytic in the unit disc : Kolmogorov theorem, Riesz' Theorem, Hardy spaces, invariant subspaces of H 2(T), Beurling's theorem. Factoring.
(e) Some applications of Fourier Transforms and Distribution theory to PDE.
(Note : it is recommended that only one of (d) or (e) is covered.)

Algebraic Topology
Homotopy, homotopy-equivalence, contractibility.
Fundamental groups. Continuous maps. Invariance under homotopy-equivalence.
Covering spaces. Computation of $\pi_1(S^1,*)$ proof of path-lifting and injectivity under $\pi_1$.
Applications. Brouwer fixed point theorem in 2 dimensions. Fundamental theorem of algebra, algebraic deductions from elementary covering space theory.

Application of Analysis to Geometry
Differentiable manifolds. Derivatives and tangents. Inverse function theorem and immersions. Submersions. Local submersion theorem. Pre-image theorem. Transversality. Homotopy and stability. Sard's theorem and its applications. Morse functions. Embedding manifolds in Euclidean space. Manifolds with boundary. Mod 2 intersection theory. Winding numbers and Jordan-Brouwer separation theorem. Borsuk-Ulam theorem. Oriented intersection theory. Lefschetz' fixed-point theory. Vector fields and Poincare-Hopf theorem. Hopf degree theorem. Euler characteristic and triangulation
(Ref. Differential Topologyby V. Guillemin and A. Pollack).

Descriptive Set Theory
Polich spaces, Cantor set, space and irrationals, transfer theorems.
Hierarchy of Borel sets, Borel functions, Borel isomorphism theorem.
Analytic sets, co-analytic sets, projective sets, separation and reduction principles, uniformisation of Boral Sets. Cardinalities and regularity properties of analytic and co-analytic sets.
Recursive functions, arithmetical and analytic point classes, codings, uniformity and good parameterizations, Recursion theorem. Pre-well ordering property. Parametrizaiton theorems, scale property, standard basis theorems, Kondo's theorem.

Advanced Probability II
Any one topic from the following :
(a) Markov process, generators. Brownian motion as a Markov process (if this topic is not covered in Advanced Probability I).
(b) Stochastic integrals and Ito calculus.
(c) Gaussian process (Ref. Kuelb).
(d) Empirical processes (Ref. Pollard/Wineller et al.).
(e) Large deviations.
(f) Strong representations (Ref. Bahadur-Kiefer, Revesz).
(g) I.D. laws, stable and semi-stable laws; semi-stable laws on R or Branch spaces.

Second-order Processes
Operators on a Hillbert space. Spectral measures. Spectral theorem, normal operators. Stone's theorem on one-parameter unitary groups.
Second-order processes, processes with orthogonal increments. Integrals with respect to processes with orthogonal increments.
Weakly stationary processes with discrete and continuous time.
Prediction problem. Deterministic and non-deterministic processes. Wiener-Kolmogorov-Szego solution of the prediction problem for discrete time. Optimal filter (both signal and observation are weakly stationary).
Introduction to stochastic integrals, Stochastic differential equations, Bucy-Kalman filter.

Topics in Mathematical Physics

(a) Classical Mechanics: Generalized coordinates, Lagrange's equation, motion in a central field, dynamics of a rigid body, general principles of mechanics. (Ref. L.D. Landau and E.M. Lifshitz : Mechanics G. Goldstein : Classical Mechanics.)
(b) Electrodynamics: Maxwell's equations, electrodynamics of material media. Plane electromagnetic wages. Theory of relativity. Relativistic dynamics. (Ref. L.D. Landau and E.M. Lifshitz : Field Theory. Bergman : Introduction to the Theory of Relativity.)
(c) Quantum Mechanics: Schrodinger equation, harmonic oscillators, operators in quantum mechanics; theory of scattering, electron, basic principles ofsecond quantization and quantum field theory. (Ref. L.D. Landau and F.M. Lifshitz : Quantum Mechanics. Powell and Craseman : Quantum Mechanics P.A.M. Dirac : The principles of Quantum Mechanics.)
(d) Statistical Physics: Ideal gas, Boltzmann statistics, Bose and Fermi statistics, Theormodynamics, King model, phase transition. (Ref. L.D. Landau and E.M. Lifshitz : Statistical Physics. R. Kube : Statistical Mechanics.)

QE Courses

Microeconomic Theory I
1. Theory of consumer behaviour : preference ordering, utility function, budget set, demand, duality theory, theory of revealed preference, aggregate demand.
2. Theory of the firm : production set, cost minimization, profit maximization, supply, duality theory, aggregate supply.
3. Equilibrium in a single market, stability, comparative statics.
4. Imperfect competition and market structure.
5. Decision-making under uncertainty : lotteries, measures of risk.

Game Theory I
(a) Non-Cooperative Games
1. Games in normal form.
2. Rationalizability and iterated deletion of never-best responses.
3. Nash equilibrium : existence, properties and applications.
4. Two-person Zero Sum Games.
5. Games in extensive form : perfect recall and behaviour strategies.
6. Credibility and Subgame Perfect Nash Equilibrium.
7. Bargaining.
8. Repeated Games; Folk Theorems.
(b) Introduction to Cooperative Games (TU games)

Econometric Methods
1. Discrete and Limited Dependent Variables Model: types of discrete choice models, linear probability model, the probit and the logit models and Tobit model.
2. Analysis of Panel Data: Fixed effects model, random effects model (error components model), fixed or random effects? – Wu-Hausman test, Swamy’s random coefficient model.
3. Specification testing and Diagnostic Checking: inferential problems in misspecified or inadequately specified models; tests based on ML principle – W, LR and Rao’s (RS) tests; White’s information matrix test; tests for non-nested hypothesis – Davidson and McKinnon’s J test and the encompassing test.
4. Simultaneous equation systems; structural and reduced forms, least squares bias problems; identification problem, estimation methods, introduction to VAR.
5. Cointegration: a general cointegrated system, two variable model: Engle-Granger method, system estimation method – Johansen procedure; error correction model and tests for cointegration; vector autoregression and Granger causality.
6. ARCH model: properties of ARCH/Garch model, different interpretations, various generalizations, estimation and testing.
7. Other methods of testing (excluding LS and ML methods): generalized method of moments (GMM) and method of least absolute deviation : basics of non-parametric regression – idea of smoothing, smoothing techniques, the kernel method and choosing the smoothing parameter.
8. Introduction to Bayesian Econometrics: Bayes’ theorem, prior probability density functions, point estimates of parameters and prediction.

Macroeconomic Theory I
1. Review of Keynes, Classics and Structuralist macroeconomics.
2. Friedman and New Classical Economics
3. New Keynesian Economics
4. Introduction to macro models of optimal behaviour over time: Ramsey-Solow and Overlapping Generations model.
5. Real Business Cycle Theory.

Microeconomic Theory II
1. General equilibrium of an exchange economy.
2. General equilibrium with production.
3. Welfare economics : the fundamental theorems of welfare economics, core of an economy, introduction to Social Choice theory
4. Asymmetric information, market failure, theory of second best and strategic interactions.
5. Introduction to non-Walrasian equilibrium.

Macroeconomic Theory II
Selected topics out of the following list :
1. Open Economy Issues
2. Overlapping Generations models: advanced topics
3. Public Debt
4. Asset Pricing
5. Optimal taxation
6. Theories of Inflation
7. Equilibrium search and matching
8. Growth and Distribution
9. Modern theories of Unemployment

Agricultural Economics
1. Growth and Fluctuations of Agricultural Output
2. Surplus Labour
3. Farm Efficiency
4. Tenurial Efficiency
5. Interlinked Factor Markets
6. Marketable Surplus
7. New Technology
8. Effect of Liberalization on Agriculture

Industrial Organization
1. Structure conduct performance paradigm.
2. Static oligopoly models, homogeneous goods, Cournot and Bertrand models, differentiated products, horizontal and vertical differentiation, models with free entry, contestable markets, Cournot and price setting, models with free entry.
3. Measures of concentration and performance.
4. Dynamic oligopoly models : entry deterrence, limit pricing, attrition and reputation models, collusion and cartels.
5. Price discrimination, price dispersion and search theory.
6. R & D and adaptation/adoption of technology : private vs. social incentives for R & D models of adoption, diffusion and transfer of technology.
7. Mergers and takeovers, firm size ad vertical integration, corporate finance.
8. Regulation of monopolies, rate of return regulation, regulation of  firms with unknown costs/demands.
9. Multinational firms.
10. Quality, durability and warranty.
11. Advertising.
12. Joint venture, licensing and patents.

Economic Development I
(a) The Dual Economy: Surplus Labour, Wage Rigidity and Unemployment
(b) Underdevelopment as a Path Dependant Process: Vicious Circles, Balanced vs. Unbalanced Growth and Big Push Theory.
(c) Growth, Development and Income Distribution
(d) Rural Markets and Institutions

Modern Growth Theory
1. Review of traditional growth models, efficiency results, barriersto growth, technical progress.
2. AK models of growth – alternative foundation.
3. Education and growth.
4. Market structure and innovation.
5. Obsolescence, Schumpeterian growth.
6. Distribution and Political Economy of growth.
7. Open growing economies, trade policies.

Social Choice and Political Economy
Selected topics from the following :
1. Classical aggregation theory : Arrow’s theorem, Harsanyi’s theorem, aggregation with rich informational structures.
2. Stochastic Dominance, Lorenz and Generalized Lorenz  orderings, Ethical approaches to measurement of inequality and poverty.
3. Classical voting theory : the Gibbard-Satterthwaite theorem, results on restricted domains, the median voter result, stochastic outcome functions.
4. The theory of implementation in complete and incomplete information settings.
5. The theory of elections, legislatures and agenda control.
6. The theory of interest groups : lobbying, bureaucracies, endogenous coalition formation.
7. Models of corruption, political economy of the state.

Incentives and Organizations
1. Theory of incentives : adverse selection, moral hazard, multiple agents, contract dynamics.
2. Organization theory : team theory, message space size, costly information processing models.
3. Incentive-based approaches : supervision, managerial slack, limited commitment.
4. Applications to the theory of the firm : decentralization, hierarchies, transfer pricing, managerial compensation, cost allocation.

Privatization and Regulations
1. Regulation of competition, externalities and natural monopolies, vertical integration, mergers and takeovers, bureaucracies and corruption.
2. Public sector performance in India and other developing countries.
3. Privatization, theory and experiences.

Economic Development II
1. Economic Development and Planning of Dual Economics : Choice of Techniques, Marketable Surplus, Rural-Urban Migration, Unemployment.
2. Role of trade and factor mobility in economic development; international technology transfer and relative technological backwardness of less developed countries.
3. Endogenous growth : increasing returns and technological progress; multiple equilibria and underdevelopment trap.

Econometric Applications II
Some subset of the following topics will be covered depending upon the interest of the instructor and the students.
1. Income and allied size distributions : Stochastic models of income  distribution, Measurement of income inequality, problems of measurement, Indian studies on inequality and poverty.
2. Advanced demand analysis : Demand systems, zero expenditure and corner solutions, nonlinear budget frontiers, rationing, sources of dynamics inconsumer behaviour, durable goods, non-parametric demand analysis.
3. Production analysis : Frontier production function, measurement of productivity and technical change, flexible forms, aggregation, properties and estimation of multi-output production and cost functions.
4. Application of Econometrics to Macro-Economic Problems : Macro-econometric models-economic issues in the specification and estimation, illustrative applications, uses in forecasting and policy evaluation.
5. Estimation of structural models of firm behaviour : Dynamic programming models, policy effects on productivity, capital formation and product-mix of firms, models of firm heterogeneity – measurement of product quality and efficiency differences among firms.
6. Empirical models of the labour market : Duration analysis, labour supply and labour demand functions including the impact of unionisation, studies on the Indian labour market.

Game Theory II
(a) Games of Incomplete Information
1. Bayes-Nash equilibrium.
2. Applications to industrial organization.
3. Reputation models.
(b) Auction theory
1. First and second price auctions.
2. The Revenue Equivalence Theorem.
3. Revenue optimal auctions in the independent values case.
4. Efficient auctions in the common-values case.
(c) The theory of equilibrium selection
1. Sequential and trembling hand perfect equilibria.
2. Forward induction.
(d) Mechanism Design
1. Strategy-proof mechanisms: the Gibbard-Satterthwaite Theorem.
2. Transferable utility and Groves-Clarke theory.
3. Bayesian Incentive compatibility.
(e) Topics in evolutionary game theory
(f) Advanced topics in cooperative games

Bayesian Econometrics
1. Principles of Bayesian analysis.
2. Simple univariate normal linear regression models.
3. Analysis of single equation nonlinear models.
4. Multivariate regression models.
5. Comparison and testing of hypothesis.
6. Simultaneous equations econometric models.

Intertemporal Economics
1. Models of intertemporal accumulation.
2. Efficient programmes, characterizations of efficiency, efficiency and present value maximization.
3. Optimal programmes, optimality criteria in discounted and undiscounted models, existence of optimal programmes.

Theory of Planning
1. Political economy of the state, alternative viewpoints.
2. Modeling government behaviour, rational choice models, median voter model, legislatures and special interest groups, bureaucracy models.
3. Planning models, centralized planning, informationally decentralized planning processes, Lange-Lerner, MDP procedures, team theory.
4. Incentives within the public sector.
5. Performance incentives for managers, decentralized organization of production, multidivisional firms, cost centres and profit centres, cost allocation transfer pricing, labour policies : Soviet and East European firms.
6. Cost-benefit analysis.
7. Pricing public sector outputs, marginal cost and average cost pricing, peak load pricing, priority pricing.

Social Accounting
1. The economic process and various concepts.
2. A system of social/national accounts.
3. National accounts and various estimates.
4. ‘Real’ gross domestic product and ‘real’ national income.
5. Estimation of national income in India.
6. Preparation of an input-output (IO) table.

Public Economics
1. Welfare objectives of the State : interpersonal utility comparisons.
2. Principles of taxation.
3. Theory of Second Best, problems of externalities & public goods.
4. Incentives and mechanism design, Gibbard-Satterthwaite theorem.
5. Tax incidence in static (partial and general equilibrium) models.
6. Tax incidence in Dynamic Models.
7. Optimal taxation and public production.
8. Dynamics, incidence and efficiency analysis of taxes.
9. Economics of corruption.
10. Economics of  Public Sector Enterprises.
11. Procurement policies : incentive contracts and auction theory.
12. Regulation of private firms.

Regional Economics
1. Introduction to regional planning.
2. Review of the Indian situation.
3. Concepts and techniques used in regional planning.
4. Regional decision making and regional balance.
5. Functional spatial configuration and regional synthesis.

International Economics I
1. Various comparative-advantage based competitive theories of international trade including the Ricardian model, the Heckscher-Ohlin model and the sector specific model and their generalizations.
2. Theory of commercial policy, tariffs, taxes and quantitative restrictions in traditional trade models.
3. Imperfectly competitive models and intra-industry trade models of international trade.
4. Trade, growth and development.
5. International factor movements.

International Economics II
1. Dynamics of Small Open Economies in Infinite Horizon and Overlapping Generations Models.
2. Non traded goods, Real Exchange Rate and the Terms of Trade.
3. Uncertainty and International Financial Markets.
4. Money and Exchange rates under flexible and fixed prices.
5. Sovereign Debt.

Advanced Topics in International Economics
1. Political economy of trade policy.
2. International trade and endogenous growth.
3. Trade and environment.
4. Trade and distribution.
5. Exchange rate dynamics in a small country setting.
6. Agency problems and international lending.
7. The New-Keynesian Models of the Open Economy.
8. International Capital Mobility and Development.

Mathematical Programming with Applications to Economics
1. Static Linear and Non-linear Programming Problems
2.   Dynamic Problems: Calculus of Variations, Optimal Control Theory and Dynamic Programming

Monetary Economics
1. Transaction, precautionary and speculative demands for money.
2. Money in an overlapping generation model, general equilibrium Baumol-Tobin model, cash-in-advance model.
3. Currency and credit with long lived agents in overlapping generations set-up.
4. Monetary policy, (non-) neutrality.
5. Money, inflation and stability, money vs. interest rate targeting.

History of Economic Thought
1. Introduction  –  relevance of the subject, the idea of a mainstream.
2. Mercantilism –  economic and political background, issues and doctrines.
3. The physiocratic breakthrough  –  focus upon production, the framework of reproduction and concept of ‘produit net’, ‘tableau economique’ and the concept of circular flow, the physiocratic system.
4. Classical political economy (CPE)  –  Adam Smith’s break and continuity with mercantilism; the physiocratic input:  transformation of the framework of reproduction through the motion of ‘stock’; the framework of value, distribution and accumulation;  the idea of free competition: price formation through equalization of rates of profit– natural rule and market price;  Ricardo’s “elimination” of  rent; the Ricardian system and its evolution through time.
5. Marx and the Marxist tradition – the ‘labour’ standpoint : view of history, concept of ‘surplus’ and class analysis.
6. The marginalist  revolution – unresolved problems of CPE; fresh search for ‘first principles’; unification of different branches economic theory under marginal calculations and demand – supply analysis.
7. The Walrasian tradition – the idea of a ‘general equilibrium’, mathematical development : connection with optimization, the ‘welfare’ branch.
8. The Marshallian tradition – the idea of a ‘short period’, theory of the firm and market structure, the Keynesian  breakthrough  –  re-emergence of  macro analysis, macro-micro relations.
9. The Mengerian tradition – subjectivism and methodological individualism, ‘new institutional economics’.
10. The ‘present’ as history – any mainstream ?

Environmental Economics
1. Theories of externalities and public goods.
2. Trade and environmental policy.
3. Design of environmental policy.
4. Marketable pollution permits.
5. Choice between permits and taxes.
6. Methods of measuring the benefits of environmental improvements.
7. Models of resource depletion, exhaustible and renewable resources.

Theory of Finance I (see AS syllabi)

Theory of Finance II (see AS syllabi)

Theory of Finance III
Advanced Topics in
1. Banking Finance
2.Market Microstructure
3. Regulation and Incentives

Political Economy and Comparative Systems
1. Classical political economy : Crystallization of the concept of “social structure” in the concept of “class”, class division and boundary of production (“productive” vs “unproductive” class/labour) in Quesnay and Smith, the systems of social accounting policy aspects, reaction against “mercantilism : theoretical structure of classical political economy, value, distribution and accumulations, the Ricardian system, the post Ricardian scene, emergence of “socialist” doctrines.
2. Marxian political economy : the broader perspectives and view of history, “modes of production” (feudalism, capitalism and socialism),the political economy of capitalism, surplus value, theories of crises.
3. Further developments in the political economy of capitalism : developments within a “class” framework, Kalecki’s theory of effective demand and business cycles, abandoning the “class” framework or the turning point in the history of economic thought, birth of “welfare economics”, “competition” and “monopoly”, Keynes’ theory of effective demand and its link up with the theory of growth.
4. Political economy of socialism : doctrines and experiences.
5. Political economy of LDCs : the intrinisic heterogeneity and amorphousness of LDCs, the “goal” of development in a historical perspective, the concept of “dual economy”, global perspectives.b