Elements of Distribution Theory

Description

This detailed introduction to distribution theory is designed as a text for the probability portion of the first year statistical theory sequence for Master's and PhD students in statistics, biostatistics, and econometrics. The text uses no measure theory, requiring only a background in calculus and linear algebra. Topics range from the basic distribution and density functions, expectation, conditioning, characteristic functions, cumulants, convergence in distribution and the central limit theorem to more advanced concepts such as exchangeability, models with a group structure, asymptotic approximations to integrals and orthogonal polynomials. An appendix gives a detailed summary of the mathematical definitions and results that are used in the book. Cover Half-title Series-title Title Copyright 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