Elements of Distribution Theory

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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