Mathematics and 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 decision 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 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 3rdLemma Asymptotic confidence intervals and tests Maximum likelihood Likelihood ratio tests . Complexity regularization(to be written) 7 Literature
Mathematics and 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 decision
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
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
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 3rdLemma
Asymptotic confidence intervals and tests
Maximum likelihood
Likelihood ratio tests .
Complexity regularization(to be written)
7 Literature
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