A Course in Large Sample Theory

Thomas S. Ferguson

• Publisher: Routledge
• Year: 2017
• eBook ISBN: 9781351470056
• Edition: 1
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

A Course in Large Sample Theory is presented in four parts. The first treats basic probabilistic notions, the second features the basic statistical tools for expanding the theory, the third contains special topics as applications of the general theory, and the fourth covers more standard statistical topics. Nearly all topics are covered in their multivariate setting.The book is intended as a first year graduate course in large sample theory for statisticians. It has been used by graduate students in statistics, biostatistics, mathematics, and related fields. Throughout the book there are many examples and exercises with solutions. It is an ideal text for self study.

1

Basic Probability Theory

1

Modes of Convergence

We begin by studying the relationships among four distinct modes of convergence of a sequence of random vectors to a limit. All convergences are defined for d-dimensional random vectors. For a random vector X = (X₁,…, X_d) ∈ ℝ^d the distribution function of X, defined for x = (x₁,…, x_d) ∈ ℝ^d is denoted by F_x(x) = P(X ≤ x) = P(X₁ ≤ x₁,…, X_d ≤ x_d). The Euclidean norm of x = (x₁,…,x_d) ∈ ℝ^d is denoted by |x| = ${(x_1^2+\cdots +x_d^2)}^{1/2}$. Let X, X₁, X₂,… be random vectors with values in ℝ^d.

DEFINITION 1. X_n converges in law to X, X_n $\stackrel{ℒ}{\to }$ X, if $F_{{\text{x}}_n}$ (x) → F_x(x) as n → ∞, for all points x at which F_x(x) is continuous.

Convergence in law is the mode of convergence most used in the following chapters. It is the mode found in the Central Limit Theorem and is sometimes called convergence in distribution, or weak convergence.

EXAMPLE 1. We say that a random vector X ∈ ℝ^d is degenerate at a point c ∈ ℝ^d if P(X = c) = 1. Let X_n ∈ ℝ¹ be degenerate at the point 1/n, for n = 1, 2,… and let X ∈ ℝ¹ be degenerate at 0. Since 1 /n converges to zero as n tends to infinity, it may be expected that $X_n\stackrel{ℒ}{\to }X$. This may be seen by checking Definition 1. The distribution function of X_n is $F_{{\text{x}}_n}$(x) = I_{[1/n, ∞)}(x), and that of X ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Part 1 Basic Probability
• Part 2 Basic Statistical Large Sample Theory
• Part 3 Special Topics
• Part 4 Efficient Estimation and Testing
• Appendix: Solutions to the exercises
• References
• Index