#### Description

This course introduces some of the key ideas of mathematical statistics related to the good performance and optimality of statistical procedures (parameter estimates and hypothesis tests) and covers many examples. The main objective is to learn how data arising from probability distributions with some unknown parameters can be used to narrow down or draw inferences about those unknown parameters. Students will learn how to construct optimal tests and estimators in many settings, particularly those involving normally distributed or large-sample data. Also listed as SURV 420.

STAT 410.

#### Topics

Probability Review:

Densities, change-of-variable, expectation, moment generating functions, conditional expectation and variance, best mean-squared-error predictors. (Optional: multivariate normal distribution) (1.5-2 weeks)

Limit Theorems:

Central Limit Theorem (plus optional supplementary discussion of multivariate CLT). 'Delta method'. (1.5 weeks)

Sampling Distributions Related to the Normal:

Distributions of sample mean and variance; x2 , t, F distributions. (Supplementary material on limiting distribution of Pearson chi-squared goodness-of-fit distribution) (2 weeks)

Estimation:

Problem of point estimation. Likelihood. Method of moments and maximum likelihood estimators (MLE's). Cramer-Rao bound, Fisher information. Asymptotic normal distributions of moments estimators and MLE's. Large-sample Confidence Intervals based upon moments and ML estimators. Relative efficiency. confidence intervals for mean, variance and two-sample parameters for normal distributions. (3.5 weeks)

Exponential families and sufficient statistics:

Definition of exponential family and sufficient statistics. Factorization theorem. Completeness, Rao-Blackwell Theorem. (1.5-2 weeks)

Hypothesis testing:

Definitions and formulation of Neyman-Pearson theory. Duality between tests and confidence intervals. Optimal (Neyman-Pearson) simple vs. simple rejection regions. Power functions. UMP tests. P-value. Generalized likelihood ratio tests and examples. (3 weeks)

Miscellaneous Topics:

Material chosen from among: order and rank statistics; linear regression; data analysis; simulation and bootstrap methods; Bayesian procedures. ( < 2 weeks)