## Syllabus For The Subject Mathematics & Statistics

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

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)

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 3rd Lemma

Asymptotic confidence intervals and tests .

Maximum likelihood .

Likelihood ratio tests .

Complexity regularization(to be written)

7 Literature