# Statistical Inference

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2 STUDENTS

Introduction and Motivation – Basic concepts of point estimation: unbiasedness, consistency and efficiency of estimators, examples – Finding Estimators: method of moments and maximum likelihood estimators, properties of maximum likelihood estimators, problems – Lower Bounds for the Variance: Frechet-Rao-Cramer, Bhattacharya, Chapman-Robbins-Kiefer inequalities, generalization of Frechet-Rao-Cramer to higher dimensions, problems – 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 – Invariance: Best equivariant estimators, problems – Bayes and Minimax Estimation: Concepts and applications – Testing of Hypotheses: Basic concepts, simple and composite hypotheses, critical region, types of error, most powerful test, Neyman-Pearson Lemma, applications – Tests for Composite Hypotheses: Families with monotone likelihood ratio, uniformly most powerful tests, applications – Unbiasedness: Unbiased tests, similarity and completeness, UMP unbiased tests – Likelihood Ratio Tests – applications to one sample and two sample problems – Invariant Tests – Contingency Tables & Chi-square tests – Wald’s sequential probability ratio test – Interval estimation: methods for finding confidence intervals, shortest length confidence intervals, problems

### Course Curriculum

 Introduction and Motivation Details 58:18 Basic Concepts of Point Estimations – I Details 59:28 Basic Concepts of Point Estimations – II Details 55:56 Finding Estimators – I Details 0:53 Finding Estimators – II Details 57:36 Finding Estimators – III Details 59:28 Properties of MLEs Details 59:21 Lower Bounds for Variance – I Details 59:8 Lower Bounds for Variance – II Details 56:35 Lower Bounds for Variance – III Details 53:35 Lower Bounds for Variance – IV Details 1:6 Sufficiency Details 58:36 Sufficiency and Information Details 55:30 Minimal Sufficiency, Completeness Details 59:48 UMVU Estimation, Ancillarity Details 59:5 Invariance – I Details 58:57 Invariance – II Details 58:24 Bayes and Minimax Estimation – I Details 57:53 Bayes and Minimax Estimation – II Details 57:58 Bayes and Minimax Estimation – III Details 55:3 Testing of Hypotheses : Basic Concepts Details 56:42 Neyman Pearson Fundamental Lemma Details 58:30 Applications of NP lemma Details 58:43 UMP Tests Details 58:46 UMP Tests (Contd.) Details 58:35 UMP Unbiased Tests Details 56:7 UMP Unbiased Tests (Contd.) Details 51:36 UMP Unbiased Tests : Applications Details 58:42 Unbiased Tests for Normal Populations Details 53:20 Unbiased Tests for Normal Populations (Contd.) Details 1:1:18 Likelihood Ratio Tests – I Details 58:15 Likelihood Ratio Tests – II Details 53:33 Likelihood Ratio Tests – III Details 54:31 Likelihood Ratio Tests – IV Details 58:11 Invariant Tests Details 53:38 Test for Goodness of Fit Details 55:58 Sequential Procedure Details 57:50 Sequential Procedure (Contd.) Details 59:31 Confidence Intervals Details 58:41 Confidence Intervals (Contd.) Details 59:14

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