Course Co-ordinated by IIT Kharagpur
 Coordinators IIT Kharagpur

Untitled Document

Point Estimation: Parametric point estimation, unbiasedness, consistency, efficiency, method of moments and maximum likelihood, lower bounds for the variance of an estimator, Frechet-Rao-Cramer, Bhattacharya, Chapman-Robbins-Kiefer inequalities. Sufficiency, minimal sufficiency, Factorization Theorem, Rao-Blackwell Theorem, completeness, Lehmann-Scheffe Theorem, UMVUE, Basu’s Theorem, invariance, best equivariant estimators,

Testing of Hypotheses: Tests of hypotheses, simple and composite hypotheses, types of error, Neyman-Pearson Lemma, families with monotone likelihood ratio, UMP, UMP unbiased and UMP invariant tests. Likelihood ratio tests - applications to one sample and two sample problems, Chi-square tests. Wald’s sequential probability ratio test.

Interval estimation: methods for finding confidence intervals, shortest length confidence intervals.

 Module No. Topic/s Lectures 1 Introduction and Motivation 1 2 Basic concepts of point estimation: unbiasedness, consistency and efficiency of estimators, examples. 1 3 Finding Estimators: method of moments and maximum likelihood estimators, properties of maximum likelihood estimators, problems. 3 4 Lower Bounds for the Variance: Frechet-Rao-Cramer, Bhattacharya, Chapman-Robbins-Kiefer inequalities, generalization of Frechet-Rao-Cramer to higher dimensions, problems. 4 5 Data Reduction: Sufficiency, Factorization Theorem, Rao-Blackwell Theorem, minimal sufficiency, completeness, Lehmann-Scheffe Theorem, applications in deriving uniformly minimum variance estimators, Ancillary statistics, Basu’s Theorem,problems. 6 6 Invariance: Best equivariant estimators, problems. 2 7 Bayes and Minimax Estimation:  Concepts and applications. 3 8 Testing of Hypotheses: Basic concepts, simple and composite hypotheses, critical region, types of error, most powerful test, Neyman-Pearson Lemma, applications. 2 9 Tests for Composite Hypotheses: Families with monotone likelihood ratio, uniformly most powerful tests, applications. 3 10 Unbiasedness: Unbiased tests, similarity and completeness, UMP unbiased tests. 3 11 Likelihood Ratio Tests - applications to one sample and two sample problems. 3 12 Invariant Tests 2 13 Contingency Tables & Chi-square tests. 2 14 Wald’s sequential probability ratio test. 2 15 Interval estimation: methods for finding confidence intervals, shortest length confidence intervals, problems. 3

Probability Theory

1. An Introduction to Probability and Statistics by V.K. Rohatgi & A.K. Md. E. Saleh.

2. Statistical Inference by G. Casella & R.L. Berger.

3. A First Course on Parametric Inference by B.K. Kale

4. Modern Mathematical Statsitics by E.J. Dudewicz & S.N. Mishra

5. Introduction to the Theory of Statistics by A.M. Mood, F.A. Graybill and D.C. Boes

1. Theory of Point Estimation by E.L. Lehmann & G. Casella

2. Testing Statistical Hypotheses by E.L. Lehmann