Unit 1: Sufficiency principle, factorization theorem, minimal sufficiency, minimal sufficient

partition, minimal sufficient statistics, minimal sufficient statistic for exponential family,

power series family, curved exponential family, and Pitman family, completeness, bounded

completeness, ancillary statistics, Basu’s theorem and its applications.

(12L + 3T)

Unit 2: Problem of point estimation, unbiased estimators, minimum variance unbiased

estimator, Rao-Blackwell theorem and Lehmann-Scheffe theorem and their applications. A

necessary and sufficient condition for an estimator to be UMVUE, Fisher information and

information matrix, Cramer-Rao inequality, Chapman-Robbins-Kiefer bound, Bhattacharya

bounds, their applications.

(12L + 3T)

Unit 3: Maximum likelihood estimator (MLE), properties of MLE, MLE in nonregular

families, method of scoring and its applications, method of moments, method of minimum

chi-square, U-statistics for expectation and variance; it’s simple properties.

(12L + 3T)

Unit 4: The concepts of prior and posterior distributions, conjugate, Jeffrey’s and improper

priors with examples, Bayes estimation under squared error and absolute error loss functions.

(12L + 3T)

References

1. Rohatgi, V.K. and Saleh, A. K. MD. E. (2015). Introduction to Probability Theory and

Mathematical Statistics -3rd edition, John Wiley & sons.

2. Lehmann, E. L. (1983). Theory of Point Estimation - John Wiley & sons.

3. Rao, C. R.(1973). Linear Statistical Inference and its Applications, 2nd edition, Wiley.

4. Kale, B.K. and Muralidharan, K. (2015). Parametric Inference: An Introduction, Alpha

Science International Ltd.

5. Mukhopadhyay, P. (2015). Mathematical Statistics, Books and Allied (p) Ltd.

6. Dudewicz, E. J. andMishra,S. N. (1988). Modern Mathematical Statistics, John Wiley

and Sons.

7. Casella, G., and Berger, R. L. (2001). Statistical Inference, 2nd edition, Duxbury press