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Dedication

Contents

Preface

1 Properties of Probability Distributions

1.1 Introduction

1.2 Basic Framework

1.3 Random Variables

1.4 Distribution Functions

1.5 Quantile Functions

1.6 Density and Frequency Functions

1.7 Integration with Respect to a Distribution Function

1.8 Expectation

Expectation of a function of a random variable

Inequalities

1.9 Exercises

1.10 Suggestions for Further Reading

2 Conditional Distributions and Expectation

2.1 Introduction

2.2 Marginal Distributions and Independence

2.3 Conditional Distributions

2.4 Conditional Expectation

2.5 Exchangeability

2.6 Martingales

2.7 Exercises

2.8 Suggestions for Further Reading

3 Characteristic Functions

3.1 Introduction

3.2 Basic Properties

Uniqueness and inversion of characteristic functions

Characteristic function of a sum

An expansion for characteristic functions

Random vectors

3.3 Further Properties of Characteristic Functions

Symmetric distributions

Lattice distributions

3.4 Exercises

3.5 Suggestions for Further Reading

4 Moments and Cumulants

4.1 Introduction

4.2 Moments and Central Moments

Central moments

Moments of random vectors

Correlation

Covariance matrices

4.3 Laplace Transforms and Moment-Generating Functions

Laplace transforms

Moment-generating functions

Moment-generating functions for random vectors

4.4 Cumulants

Cumulants of a random vector

4.5 Moments and Cumulants of the Sample Mean

Central moments of X

4.6 Conditional Moments and Cumulants

4.7 Exercises

4.8 Suggestions for Further Reading

5 Parametric Families of Distributions

5.1 Introduction

5.2 Parameters and Identifiability

Identifiability

Likelihood ratios

5.3 Exponential Family Models

Natural parameters

Some distribution theory for exponential families

5.4 Hierarchical Models

Models for heterogeneity and dependence

5.5 Regression Models

5.6 Models with a Group Structure

Transformation models

Invariance

Equivariance

5.7 Exercises

5.8 Suggestions for Further Reading

6 Stochastic Processes

6.1 Introduction

6.2 Discrete Time Stationary Processes

6.3 Moving Average Processes

6.4 Markov Processes

Markov chains

6.5 Counting Processes

Poisson processes

Distribution of the interarrival times

6.6 Wiener Processes

Irregularity of the sample paths of a Wiener process

TheWiener process as a martingale

6.7 Exercises

6.8 Suggestions for Further Reading

7 Distribution Theory for Functions of Random Variables

7.1 Introduction

7.2 Functions of a Real-Valued Random Variable

7.3 Functions of a Random Vector

Functions of lower dimension

Functions that are not one-to-one

Application of invariance and equivariance

7.4 Sums of Random Variables

7.5 Order Statistics

Pairs of order statistics

7.6 Ranks

7.7 Monte Carlo Methods

7.8 Exercises

7.9 Suggestions for Further Reading

8 Normal Distribution Theory

8.1 Introduction

8.2 Multivariate Normal Distribution

Density of the multivariate normal distribution

8.3 Conditional Distributions

Conditioning on a degenerate random variable

8.4 Quadratic Forms

8.5 Sampling Distributions

8.6 Exercises

8.7 Suggestions for Further Reading

9 Approximation of Integrals

9.1 Introduction

9.2 Some Useful Functions

Gamma function

Incomplete gamma function

9.3 Asymptotic Expansions

Integration-by-parts

9.4 Watson’s Lemma

9.5 Laplace’s Method

9.6 Uniform Asymptotic Approximations

9.7 Approximation of Sums

9.8 Exercises

9.9 Suggestions for Further Reading

10 Orthogonal Polynomials

10.1 Introduction

10.2 General Systems of Orthogonal Polynomials

Construction of orthogonal polynomials

Zeros of orthogonal polynomials and integration

Completeness and approximation

10.3 Classical Orthogonal Polynomials

Hermite polynomials

Laguerre polynomials

10.4 Gaussian Quadrature

10.5 Exercises

10.6 Suggestions for Further Reading

11 Approximation of Probability Distributions

11.1 Introduction

11.2 Basic Properties of Convergence in Distribution

Uniformity in convergence in distribution

Convergence in distribution of random vectors

11.3 Convergence in Probability

Convergence in probability to a constant

Convergence in probability of random vectors and random matrices

11.4 Convergence in Distribution of Functions of Random Vectors

11.5 Convergence of Expected Values

11.6 Op and op Notation

11.7 Exercises

11.8 Suggestions for Further Reading

12 Central Limit Theorems

12.1 Introduction

12.2 Independent, Identically Distributed Random Variables

12.3 Triangular Arrays

12.4 Random Vectors

12.5 Random Variables with a Parametric Distribution

12.6 Dependent Random Variables

12.7 Exercises

12.8 Suggestions for Further Reading

13 Approximations to the Distributions of More General Statistics

13.1 Introduction

13.2 Nonlinear Functions of Sample Means

13.3 Order Statistics

Central order statistics

Pairs of central order statistics

Sample extremes

13.4 U-Statistics

13.5 Rank Statistics

13.6 Exercises

13.7 Suggestions for Further Reading

14 Higher-Order Asymptotic Approximations

14.1 Introduction

14.2 Edgeworth Series Approximations

Third- and higher-order approximations

Expansions for quantiles

14.3 Saddlepoint Approximations

Renormalization of saddlepoint approximations

Integration of saddlepoint approximations

14.4 Stochastic Asymptotic Expansions

14.5 Approximation of Moments

14.6 Exercises

14.7 Suggestions for Further Reading

Appendix 1 Integration with Respect to a Distribution Function

A1.1 Introduction

A1.2 A General Definition of Integration

A1.3 Convergence Properties

A1.4 Multiple Integrals

A1.5 Calculation of the Integral

A1.6 Fundamental Theorem of Calculus

A1.7 Interchanging Integration and Differentiation

Appendix 2 Basic Properties of Complex Numbers

A2.1 Definition

A2.2 Complex Exponentials

A2.3 Logarithms of Complex Numbers

Appendix 3 Some Useful Mathematical Facts

A3.1 Sets

A3.2 Sequences and Series

A3.3 Functions

A3.4 Differentiation and Integration

A3.5 Vector Spaces

References

Name Index

Subject Index