## Syllabus For The Subject Mathematics and Statistics

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

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)

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