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MATHEMATICS STATISTICS
Contents
1 Introduction
Some notation and model assumptions .
Estimation
Comparison of estimators: risk functions .
Comparison of estimators: sensitivity .
Confidence intervals
Equivalence confidence sets and tests .
Intermezzo: quantile functions
How to construct tests and confidence sets .
An illustration: the two-sample problem .
Assuming normality
A nonparametric test
Comparison of Student’s test and Wilcoxon’s test .
How to construct estimators . .
Plug-in estimators .
The method of moments .
Likelihood methods
2 Decision theory
Decisions and their risk
Admissibility . .
Minimaxity
Bayes decisions
Intermezzo: conditional distributions .
Bayes methods .
Discussion of Bayesian approach (to be written)
Integrating parameters out (to be written)
Intermezzo: some distribution theory .
The multinomial distribution .
The Poisson distribution .
The distribution of the maximum of two random variables
Sufficiency .
Rao-Blackwell .
Factorization Theorem of Neyman .
Exponential families .
Canonical form of an exponential family .
Minimal sufficiency .
3 Unbiased estimators
What is an unbiased estimator? .
UMVU estimators .
Complete statistics .
The Cramer-Rao lower bound .
Higher-dimensional extensions .
Uniformly most powerful tests .
An example .
UMP tests and exponential families
Unbiased tests
Conditional tests .
4 Equivariant statistics
Equivariance in the location model .
Equivariance in the location-scale model (to be written)
5 Proving admissibility and minimaxity
Minimaxity .
Admissibility
Inadmissibility in higher-dimensional settings (to be written)
6 Asymptotic theory
Types of convergence .
Stochastic order symbols .
Some implications of convergence .
Consistency and asymptotic normality .
Asymptotic linearity .
The δ-technique
M-estimators
Consistency of M-estimators .
Asymptotic normality of M-estimators .
Plug-in estimators
Consistency of plug-in estimators .
Asymptotic normality of plug-in estimators .
Asymptotic relative efficiency .
Asymptotic Cramer Rao lower bound
Le Cam’s 3rd Lemma
Asymptotic confidence intervals and tests .
Maximum likelihood .
Likelihood ratio tests .
Complexity regularization(to be written)
7 Literature