# Course

## M481 - Statistical Inference I

Course No:
M481
Credit:
4
Prerequisites:
M206 and M305 or equivalent courses
Approval:
2014
UG-Elective
PG-Core
Syllabus:

Review: joint and conditional distributions, order statistics, group family, exponential family (3 hrs)

Introduction to parametric inference, sufficiency principle and data reduction, factorization theorem, minimal sufficient statistics, Fisher information, ancillary statistics, complete statistics, Basu’s theorem (9 hrs)

Unbiasedness, best unbiased and linear unbiased estimator, Rao-Blackwell theorem, Lehmann-Scheffe theorem and UMVUE, Cramer-Rao lower bound and UMVUE, multi-parameter cases (8 hrs)

Location and scale invariance, principle of equivariance (4 hrs)

Methods of estimation: method of moments, likelihood principle and maximum likelihood estimation, properties of MLE: invariance, consistency, asymptotic normality (5 hrs)

Hypothesis testing: error probabilities and power, most powerful tests, Neyman-Pearson lemma and its applications, p-value, uniformly most powerful (UMP) test via Neyman-Pearson lemma, UMP test via monotone likelihood ratio property, existence and nonexistence of UMP test for two sided alternative, unbiased and UMP unbiased tests (13 hrs)

Likelihood (generalized) ratio tests and its properties, invariance and most powerful invariant tests (5 hrs)

Introduction to confidence interval estimation, methods of fining confidence intervals: pivotal quantity, inversion of a test, examples such as confidence interval for mean, variance, difference in means, optimal interval estimators, uniformly most accurate confidence bound, large sample confidence intervals (7 hrs)

Reference Books:
1. Lehmann, E.L. and Casella, G.(1998), “Theory of Point Estimation”, 2nd edition, New York: Springer.
2. Lehmann, E.L. and Romano, J. P. (2005), “Testing Statistical Hypotheses”, 3rd edition, Springer.
3. Nitis Mukhopadhyay (2000), “Probability and Statistical Inference”, New York: Marcel Dekker.
4. George Casella and Roger L. Berger, “Statistical Inference”, 2nd edition, Cengage Learning, 2001.
5. A.M. Mood, F.A. Graybill and D.C. Boes (1974), “Introduction to the Theory of Statistics”, 3rd edition, McGraw Hill.