STOR 654 Study Guide


Preliminaries

Order statistics

Stirling's formula

Convex and concave functions

Convexity based inequalities: Jensen, Holder, Cauchy Schwartz

Order based inequality: association inequality for increasing and decreasing functions

Elementary inequalities for probabilities: Markov, Chebyshev, and Chernoff

Moment generating functions



Statistical Inference

Decision theoretic framework, including loss, risk and admissibility

Location and scale families

Exponential families, including canonical exponential families

Sufficiency and the factorization theorem

Minimal sufficiency

Ancillary and complete statistics, Basu's theorem

Likelihood



Point Estimation

Method of moments

Maximum likelihood estimation, including evaluation of maxima

Bayesian point estimators, including prior and posterior distributions, and conjugate families

Unbiased estimators

Squared error loss and the bias-variance decomposition

Best unbiased estimators

The Cramer-Rao inequality and Fisher Information

Rao-Blackwell theorem

Bayes risk and Bayes rules



Hypothesis Testing

Likelihood ratio tests

Power function and Type I, II errors

Level, size and unbiasedness

Uniformly most powerful tests: Neyman-Pearson Lemma, tests based on sufficient statistics

Monotone likelihood ratio and Karlin-Rubin Theorem

p-values



Interval Estimation

Coverage probability and confidence coefficient

Inverting hypothesis tests

Confidence sets from pivots

Pivoting CDFs