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
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
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