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Theory of Statistical Inference



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ISBN 9780367488758
December 22, 2021 Forthcoming by Chapman and Hall/CRC
472 Pages 26 B/W Illustrations

 
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Book Description

The purpose of applying mathematical theory to the theory of statistical inference is to make it simpler and more elegant. Theory of Statistical Inference is concerned with the development of a type of optimization theory which can be used to inform the choice of statistical methodology. Of course, this would be pointless without reference to such methods. We are simply noting that they are included in support of the larger goal. This book distinguishes itself from other graduate textbooks because it is written from the point of view that some degree of understanding of measure theory, as well as other branches of mathematics, which include topology, group theory and complex analysis, should be a part of the canon of statistical inference.

Features

  • Focuses on mathematics underpinnings of statistics
  • Provides an analytical framework via optimization theory with which to inform the choice of statistical method
  • Uses R -Emphasizes information theory
  • Contains treatment of linear models, GLMs and Estimating Equations

Table of Contents

Preface

1 Distribution Theory

1.1 Introduction

1.2 Probability Measures

1.3 Some Important Theorems of Probability

1.4 Commonly Used Distributions

1.5 Stochastic Order Relations

1.6 Quantiles

1.7 Inversion of the CDF

1.8 Transformations of Random Variables

1.9 Moment Generating Functions

1.10 Moments and Cumulants

1.11 Problems

2 Multivariate Distributions

2.1 Introduction

2.2 Parametric Classes of Multivariate Distributions

2.3 Multivariate Transformations

2.4 Order Statistics

2.5 Quadratic Forms, Idempotent Matrices and Cochran’s Theorem

2.6 MGF and CGF of Independent Sums

2.7 Multivariate Extensions of the MGF

2.8 Problems

3 Statistical Models

3.1 Introduction

3.2 Parametric Families for Statistical Inference

3.3 Location-Scale Parameter Models

3.4 Regular Families

3.5 Fisher Information

3.6 Exponential Families

3.7 Sufficiency

3.8 Complete and Ancillary Statistics

3.9 Conditional Models and Contingency Tables

3.10 Bayesian Models

3.11 Indifference, Invariance and Bayesian Prior Distributions

3.12 Nuisance Parameters

3.13 Principles of Inference

3.14 Problems

4 Methods of Estimation

4.1 Introduction

4.2 Unbiased Estimators

4.3 Method of Moments Estimators

4.4 Sample Quantiles and Percentiles

4.5 Maximum Likelihood Estimation

4.6 Confidence Sets

4.7 Equivariant Versus Shrinkage Estimation

4.8 Bayesian Estimation

4.9 Problems

5 Hypothesis Testing

5.1 Introduction

5.2 Basic Definitions

5.3 Principles of Hypothesis Tests

5.4 The Observed Level of Significance (P-Values)

5.5 One and Two Sided Tests

5.6 Hypothesis Tests and Pivots

5.7 Likelihood Ratio Tests

5.8 Similar Tests

5.9 Problems

6 Linear Models

6.1 Introduction

6.2 Linear Models - Definition

6.3 Best Linear Unbiased Estimators (BLUE)

6.4 Least-squares Estimators, BLUEs and Projection Matrices

6.5 Ordinary and Generalized Least-Squares Estimators

6.6 ANOVA Decomposition and the F Test for Linear Models

6.7 The F Test for One-Way ANOVA

6.8 Simultaneous Confidence Intervals

6.9 Multiple Linear Regression

6.10 Problems

7 Decision Theory

7.1 Introduction

7.2 Ranking Estimators by MSE

7.3 Prediction

7.4 The Structure of Decision Theoretic Inference

7.5 Loss and Risk

7.6 Uniformly Minimum Risk Estimators (The Location-Scale Model

7.7 Some Principles of Admissibility

7.8 Admissibility for Exponential Families (Karlin’s Theorem)

7.9 Bayes Decision Rules

7.10 Admissibility and Optimality

7.11 Problems

8 Uniformly Minimum Variance Unbiased (UMVU) Estimation

8.1 Introduction

8.2 Definition of UMVUE’s

8.3 UMVUE’s and Sufficiency

8.4 Methods of Deriving UMVUEs

8.5 Nonparametric Estimation and U-statistics

8.6 Rank Based Measures of Correlation

8.7 Problems

9 Group Structure and Invariant Inference

9.1 Introduction

9.2 MRE Estimators for Location Parameters

9.3 MRE Estimators for Scale Parameters

9.4 Invariant Density Families

9.5 Some Applications of Invariance

9.6 Invariant Hypothesis Tests

9.7 Problems

10 The Neyman-Pearson Lemma

10.1 Introduction

10.2 Hypothesis Test as Decision Rules

10.3 Neyman-Pearson (NP) Tests

10.4 Monotone Likelihood Ratios (MLR)

10.5 The Generalized Neyman-Pearson Lemma

10.6 Invariant Hypothesis Tests

10.7 Permutation Invariant Tests

10.8 Problems

11 Limit Theorems

11.1 Introduction

11.2 Limits of Sequences of Random Variables

11.3 Limits of Expected Values

11.4 Uniform Integrability

11.5 The Law of Large Numbers

11.6 Weak Convergence

11.7 Multivariate Extensions of Limit Theorems

11.8 The Continuous Mapping Theorem

11.9 MGFs, CGFs and Weak Convergence

11.10 The Central Limit Theorem for Triangular Arrays

11.11 Weak Convergence of Random Vectors

11.12 Problems

12 Large Sample Estimation - Basic Principles

12.1 Introduction

12.2 The _-Method

12.3 Variance Stabilizing Transformations

12.4 The _-Method and Higher Order Approximations

12.5 The Multivariate _-Method

12.6 Approximating the Distributions of Sample Quantiles: The Bahadur Representation Theorem

12.7 A Central Limit Theorem for U-statistics

12.8 The Information Inequality

12.9 Asymptotic Efficiency

12.10 Problems

13 Asymptotic Theory for Estimating Equations

13.1 Introduction

13.2 Consistency and Asymptotic Normality of M-estimators

13.3 Asymptotic Theory of MLEs

13.4 A General Form for Regression Models

13.5 Nonlinear Regression

13.6 Generalized Linear Models (GLMs)

13.7 Generalized Estimating Equations (GEE)

13.8 Consistency of M-estimators

13.9 Asymptotic Distribution of ˆ_n

13.10 Regularity Conditions for Estimating Equations

13.11 Problems

14 Large Sample Hypothesis Testing

14.1 Introduction

14.2 Model Assumptions

14.3 Large Sample Tests for Simple Hypotheses

14.4 Nuisance Parameters and Composite Null Hypotheses

14.5 A Comparison of the LR, Wald and Score Tests

14.6 Pearson’s _2 Test for Independence in Contingency Tables

14.7 Estimating Power for Approximate _2 Tests

14.8 Problems

A Parametric Classes of Densities

B Topics in Linear Algebra

B.1 Numbers

B.2 Equivalence Relations

B.3 Vector Spaces

B.4 Matrices

B.5 Dimension of a Subset of Rd

C Topics in Real Analysis and Measure Theory

C.1 Metric spaces

C.2 Measure Theory

C.3 Integration

C.4 Exchange of Integration and Differentiation

C.5 The Gamma and Beta Functions

C.6 Stirling’s Approximation of the Factorial

C.7 The Gradient Vector and the Hessian Matrix

C.8 Normed Vector Spaces

C.9 Taylor’s Theorem

D Group Theory

D.1 Definition of a Group

D.2 Subgroups

D.3 Group Homomorphisms

D.4 Transformation Groups

D.5 Orbits and Maximal Invariants

Bibliography

Index

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Author(s)

Biography

Anthony Almudevar is Associate Professor of Biostatistics and Computational Biology at the University of Rochester.