Statistical Inference-I

Point estimation, criteria of a good estimator: unbiasedness, consistency, efficiency and sufficiency. Concept of sufficient statistics, Fisher Neyman factorization theorem, Cramer-Rao inequality, Bhattacharya Bounds, Rao-Blackwell theorem, Completeness and Lehmann-Scheffe theorem, Uniformly minimum variance unbiased estimator, minimal sufficient statistic.

Methods of Estimation: Maximum likelihood method, moments, minimum Chi-square and modified minimum Chi-square methods. Properties of maximum likelihood estimator (without proof). Confidence intervals: Determination of confidence intervals based on large samples, confidence intervals based on small samples.

Statistical Hypothesis: Simple and composite, procedure of testing of hypothesis, critical region, types of errors, level of significance, p-value, power of a test, most powerful test and Neyman-Pearson fundamental lemma.

Sequential Analysis: Definition and construction of S.P.R.T. Fundamental relation among, A and B. Wald’s inequality for testing null hypothesis v/s alternative hypothesis. Determination of A and B Average sample number and operating characteristic curve, and determination of OC and ASN functions through Wald’s fundamental identity.

Non-Parametric Tests: Sign tests, signed rank test, Kolmogorov-Smirnov one sample test. General two sample problems: Wolfowitz runs test, Kolmogorov Smirnov two sample test (for sample of equal size), Median test, Wilcoxon-Mann-Whitney U test. Test of randomness using run test based on the total number of runs and the length of a run. Kendall’s Tau test for independence of correlation, Kruskal Wallis K sample test and concept of asymptotic relative efficiency(ARE).