Unit 1: Consistency of an estimator, weak and strong consistency, joint and marginal consistency, invariance property under continuous transformations, methods of constructing consistent estimators, asymptotic relative efficiency. Consistent and Asymptotic Normal (CAN) Estimators: Definition of CAN estimator for real and vector valued parameters, invariance of CAN property under non-vanishing differentiable transformation. Methods of constructing CAN estimators: Method of Moments, method of percentiles, comparison of CAN estimators. (12+ 3T)
Unit 2: CAN and BAN estimators in one parameter and multi-parameter exponential family of
distributions, BAN estimators, super efficient estimators, Crammer regularity conditions, Cramer – Huzurbazar results.
(12L+3T)
Unit 3: Variance stabilizing transformations; their existence; their applications in obtaining large sample tests and estimators. Asymptotic Confidence Intervals based on CAN estimators and based on VST, Asymptotic Confidence regions in multi-parameter families. (12L+3T)
Unit 4: Likelihood ratio test and its asymptotic distribution, Wald test, Rao’s Score test, Pearson Chi-square test for goodness of fit, Bartlett’s test for homogeneity of variances. Consistent test, comparison of tests: asymptotic relative efficiency of tests (Pitman and Bahadur efficiency). Performance evaluation (based on simulation) of asymptotic tests and confidence intervals.
(12L +3T)
References:
i. Kale B.K. (1999): A first course on parametric inference, Narosa Pub.
ii. Zacks S. (1971): Theory of statistical inference, Wiley & Sons inc.
iii. Rohatagi V.K. and Saleh A. K. Md. E.(2001) : Introduction to Probability Theory and Mathematical Statistics- John Wiley and sons Inc.
iv. Ferguson, T.S. (1996): A Course in Large Sample Theory. Chapman and Hall
v. Lehmann E L (1999): Elements of Large Sample Theory, Springer.
DasGupta A. (2008): Asymptotic Theory of Statistics and Probability, Springer Texts in Statistics.- Teacher: S.B. Mahadik