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Course Co-ordinated by IIT Kharagpur
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Prof. Somesh Kumar
IIT Kharagpur

 

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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



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