Statistical Inference

Overview

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, Basus 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 - Walds sequential probability ratio test - Interval estimation: methods for finding confidence intervals, shortest length confidence intervals, problems