Statistical Inference–II
M.Sc. Statistics · Semester III · Academic Session 2026–27
The course gives a unified treatment of statistical decision theory, sequential procedures and distribution-free methods: how loss and risk functions provide a common framework for estimation and testing, how the sample size can be decided sequentially through an SPRT, and how valid inference can be made when no fully specified parametric model is available.
Slides
There is one deck for each lecture.
Decision Theory · Lectures 1–20
| Lecture | Topics | Slides |
|---|---|---|
| 1 | Introduction; elements of a decision problem; standard loss functions | 19 slides |
| 2 | Non-randomised decision rules; the risk function; risk of an estimator and of a test | 8 slides |
| 3 | Randomised decision rules; convexity of the risk set | 10 slides |
| 4 | Uniformly best rules; admissibility; linear estimators of a normal mean | 13 slides |
| 5 | The minimax principle; minimax action in a no-data problem | 9 slides |
| 6 | Minimax and Bayes rules among non-randomised rules: the Bernoulli example | 10 slides |
| 7 | Prior distribution, Bayes risk and Bayes rule | 10 slides |
| 8 | Structure of the Bayes rule: posterior expected loss | 7 slides |
| 9 | Bayes estimators under squared error, weighted squared error and absolute error loss | 11 slides |
| 10 | Small-neighbourhood (MAP) and LINEX losses | 10 slides |
| 11 | Worked examples: linear rules; the beta–binomial model | 18 slides |
| 12 | Normal–normal and Poisson–gamma models; different losses, different estimates | 21 slides |
| 13 | Bayes rules and admissibility | 11 slides |
| 14 | Least favourable priors; Bayes rules with constant risk are minimax | 10 slides |
| 15 | Randomised minimax rules in finite problems | 14 slides |
| 16 | Minimax estimators of a binomial proportion and of a normal mean | 15 slides |
| 17 | Complete, essentially complete and minimal complete classes and their relationship | 12 slides |
| 18 | Complete class theorems; sufficiency; estimation viewed as a decision problem | 14 slides |
| 19 | Testing viewed as a decision problem; the Bayes test | 14 slides |
| 20 | The minimax test; summary and revision | 17 slides |
Sequential Analysis · Lectures 21–40
| Lecture | Topics | Slides |
|---|---|---|
| 21 | Sequential procedures; stopping rules and stopping times | 6 slides |
| 22 | Wald’s SPRT; the log-likelihood ratio as a random walk | 7 slides |
| 23 | Termination of the SPRT with probability one | 7 slides |
| 24 | Choice of the boundaries; Wald’s approximations | 7 slides |
| 25 | Wald’s bounds on the attained error probabilities; the SPRT in terms of a sufficient statistic | 10 slides |
| 26 | Wald’s fundamental identity for bounded stopping times | 7 slides |
| 27 | The fundamental identity for the SPRT; Wald’s equation | 8 slides |
| 28 | Wald’s second identity | 6 slides |
| 29 | The O.C. function: the root \(h(\theta)\) and the O.C. formula | 11 slides |
| 30 | The parametric method for drawing the O.C. curve | 6 slides |
| 31 | The A.S.N. function | 8 slides |
| 32 | The zero-drift case | 5 slides |
| 33 | SPRT for Bernoulli and binomial populations | 12 slides |
| 34 | SPRT for a Poisson population | 6 slides |
| 35 | SPRT for a normal mean and a normal variance | 9 slides |
| 36 | SPRT for exponential and gamma populations | 12 slides |
| 37 | O.C. and A.S.N. curves; comparison with a fixed-sample test | 14 slides |
| 38 | Solved numerical examples: binomial, Poisson and normal | 14 slides |
| 39 | Solved numerical examples: exponential and gamma | 10 slides |
| 40 | Summary and revision | 10 slides |
Nonparametric Inference · Lectures 41–60
| Lecture | Topics | Slides |
|---|---|---|
| 41 | Nonparametric and distribution-free procedures; assumptions that still matter | 9 slides |
| 42 | Empirical distribution function; probability integral transformation | 10 slides |
| 43 | Order statistics; the distribution of \(X_{(r)}\) | 7 slides |
| 44 | Sample quantiles: consistency and asymptotic normality | 9 slides |
| 45 | Distribution-free confidence intervals for quantiles | 9 slides |
| 46 | One-sided bounds; duality with the sign test; principles of nonparametric testing | 9 slides |
| 47 | Runs: the null distribution of the number of runs | 7 slides |
| 48 | Mean and variance of the number of runs; the one-sample runs test | 15 slides |
| 49 | Runs up and down; the Wald–Wolfowitz two-sample runs test | 12 slides |
| 50 | One-sample sign test | 7 slides |
| 51 | Paired sign test; confidence interval for a median | 9 slides |
| 52 | Median test: construction; a \(k\)-sample example | 11 slides |
| 53 | Median test for two samples; Fisher’s exact test | 8 slides |
| 54 | Wilcoxon signed-rank test: null distribution | 12 slides |
| 55 | Paired signed-rank test with ties; Hodges–Lehmann estimator | 13 slides |
| 56 | Mann–Whitney–Wilcoxon test: null distribution | 12 slides |
| 57 | Mann–Whitney test with ties; normal approximation; Hodges–Lehmann shift estimator | 16 slides |
| 58 | One-sample Kolmogorov–Smirnov test | 17 slides |
| 59 | Two-sample Kolmogorov–Smirnov test | 8 slides |
| 60 | Comparison and selection of procedures; summary and revision | 13 slides |
Syllabus
| Unit | Topics | Lectures |
|---|---|---|
| I | Decision theory: loss and risk functions; randomised and non-randomised decision rules; admissibility; complete, essentially complete and minimal complete classes; Bayes and minimax decision rules; estimation and testing viewed as decision problems; Bayes and minimax estimators; Bayes and minimax tests in simple cases. | 20 |
| II–III | Sequential analysis: SPRT and its applications to binomial, Poisson, normal, exponential and gamma populations; O.C. and A.S.N. functions; termination of the SPRT with probability one; Wald’s fundamental identity and its applications. | 20 |
| IV–V | Nonparametric inference: probability integral transformation; estimation of quantiles and distribution-free confidence intervals for population quantiles; tests for randomness; runs and sign tests (one and two samples); median test; Wilcoxon and Mann–Whitney tests; one- and two-sample Kolmogorov–Smirnov tests. | 20 |
Books
- Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd Ed., Springer-Verlag, New York.
- Conover, W. J. (1999). Practical Nonparametric Statistics, 3rd Ed., John Wiley & Sons, New York.
- Ferguson, T. S. (1967). Mathematical Statistics: A Decision Theoretic Approach, Academic Press, New York.
- Gibbons, J. D., and Chakraborti, S. (2021). Nonparametric Statistical Inference, 6th Ed., CRC Press, Boca Raton.
- Goon, A. M., Gupta, M. K., and Dasgupta, B. (1980). An Outline of Statistical Theory, Volume II, 2nd revised Ed., The World Press, Calcutta.
- Lehmann, E. L., and Romano, J. P. (2022). Testing Statistical Hypotheses, 4th Ed., Springer, Cham.
- Rohatgi, V. K., and Saleh, A. K. Md. E. (2015). An Introduction to Probability and Statistics, 3rd Ed., John Wiley & Sons, Hoboken.
- Wald, A. (1947). Sequential Analysis, John Wiley & Sons, New York.
The slides follow my lecture notes for the course. The complete notes, worked solutions and assignments are shared with the students enrolled in the course.