730 Lecture 22 Todays lecture 10222021 730 lecture

730 Lecture 22 Today’s lecture: 10/22/2021 730 lecture 22 1

Basic Stuff • Distributions • Distribution functions • Moments and inequalities – Jensen – Chebychev • Joint/marginal/conditional • Convergence • Order statistics 10/22/2021 730 lecture 22 2

Estimation • Sampling distributions – Standard errors • Properties of estimators – – – 10/22/2021 Unbiasedness MSE CR lower bound Sufficiency Minimal sufficiency Exponential families 730 lecture 22 3

Estimation II • UMVUES – Recognizing using CR lower bound, score – Construction using Rao-Blackwellization • Likelihood – MLE’s – Calculation of MLE’s – Numerical methods • Fisher scoring, Newton-Raphson, EM, IRLS – Generalized linear models 10/22/2021 730 lecture 22 4

Confidence intervals • • Inverting tests Pivots Examples Average length and coverage 10/22/2021 730 lecture 22 5

Asymptotic Theory • Expansions for expected values, bias, variance, MSE based on Taylor series • Approximation of sampling distributions using convergence in distribution, CLT • Asymptotic distribution of MLE • Asymptotic distribution of LR, score and Wald tests – Simple hypothesis (parameter specified) – Compound hypothesis (only part specified) 10/22/2021 730 lecture 22 6

Computer-intensive methods • Bootstrap estimates • Bootstrap confidence intervals – Percentile-t intervals 10/22/2021 730 lecture 22 7

LR/Wald/Score: graph of log -likelihood 10/22/2021 730 lecture 22 8