Examples and Problems in Mathematical Statistics (eBook)

Shelemyahu Zacks (Autor)

eBook Download: PDF | EPUB

2013 | 1. Auflage
654 Seiten
Wiley (Verlag)
9781118605837 (ISBN)

Infos

This book presents examples that illustrate the theory of mathematical statistics and details how to apply the methods for solving problems.  While other books on the topic contain problems and exercises, they do not focus on problem solving. This book fills an important niche in the statistical theory literature by providing a theory/example/problem approach.  Each chapter is divided into four parts: Part I provides the needed theory so readers can become familiar with the concepts, notations, and proven results; Part II presents examples from a variety of fields including engineering, mathematics, and statistics; Part III contains the problems for solutions; and Part IV features selected problems with solutions.  Within the book's nine chapters, the authors provides 200 examples and over 300 problems, and solutions are provided for approximately 10% of the problems.  Chapter coverage includes: Basic Probability Theory; Statistical Distributions; Sufficient Statistics and Information in Samples; Testing Statistical Hypothesis; Statistical Estimation; Confidence and Tolerance Intervals; Large Sample Theory for Estimation and Testing; Bayesian Analysis in Testing and Estimation; and Advanced Topics in Estimation Theory.

Autor

SHELEMYAHU ZACKS, PHD, is Distinguished Professor in the Department of Mathematical Sciences at Binghamton University. He has published several books and more than 170 journal articles on the design and analysis of experiments, statistical control of stochastic processes, statistical decision theory, sequential analysis, reliability, statistical methods in logistics, and sampling from finite populations. A Fellow of the American Statistical Association, Institute of Mathematical Sciences, and American Association for the Advancement of Sciences, Dr. Zacks is the author of Stage& #45;Wise Adaptive Designs, also published by Wiley.

Textauszug

CHAPTER 1

Basic Probability Theory

PART I: THEORY

It is assumed that the reader has had a course in elementary probability. In this chapter we discuss more advanced material, which is required for further developments.

1.1 OPERATIONS ON SETS

Let denote a sample space. Let E1, E2 be subsets of . We denote the union by E1 E2 and the intersection by E1 E2. = − E denotes the complement of E. By DeMorgan’s laws = 1 2 and = 1 2.

Given a sequence of sets {En, n ≥ 1} (finite or infinite), we define

(1.1.1)

Furthermore, and are defined as

(1.1.2)

If a point of belongs to En, it belongs to infinitely many sets En. The sets , En and , En always exist and

(1.1.3)

If , En = , En, we say that a limit of {En, n ≥ 1} exists. In this case,

(1.1.4)

A sequence {En, n ≥ 1} is called monotone increasing if En En+1 for all n ≥ 1. In this case . The sequence is monotone decreasing if En En+1, for all n ≥ 1. In this case . We conclude this section with the definition of a partition of the sample space. A collection of sets = {E1, …, Ek} is called a finite partition of if all elements of are pairwise disjoint and their union is , i.e., Ei Ej = for all i ≠ j; Ei, Ej ; and . If contains a countable number of sets that are mutually exclusive and , we say that is a countable partition.

1.2 ALGEBRA AND σ–FIELDS

Let be a sample space. An algebra is a collection of subsets of satisfying

(1.2.1)

We consider = . Thus, (i) and (ii) imply that . Also, if E1, E2 then E1 E2 .

The trivial algebra is 0 = {, }. An algebra 1 is a subalgebra of 2 if all sets of 1 are contained in 2. We denote this inclusion by 1 2. Thus, the trivial algebra 0 is a subalgebra of every algebra . We will denote by (), the algebra generated by all subsets of (see Example 1.1).

If a sample space has a finite number of points n, say 1 ≤ n < ∞, then the collection of all subsets of is called the discrete algebra generated by the elementary events of . It contains 2n events.

Let be a partition of having k, 2 ≤ k, disjoint sets. Then, the algebra generated by , (), is the algebra containing all the 2k − 1 unions of the elements of and the empty set.

An algebra on is called a σ–field if, in addition to being an algebra, the following holds.

(iv) If En , n ≥ 1, then En .

We will denote a σ–field by . In a σ–field the supremum, infinum, limsup, and liminf of any sequence of events belong to . If is finite, the discrete algebra () is a σ–field. In Example 1.3 we show an algebra that is not a σ–field.

The minimal σ–field containing the algebra generated by {(-∞, x], -∞ < x < ∞ } is called the Borel σ–field on the real line .

A sample space , with a σ–field , (, ) is called a measurable space.

The following lemmas establish the existence of smallest σ–field containing a given collection of sets.

Lemma 1.2.1 Let be a collection of subsets of a sample space . Then, there exists a smallest σ–field (), containing the elements of .

Proof.   The algebra of all subsets of , () obviously contains all elements of . Similarly, the σ–field containing all subsets of , contains all elements of . Define the σ–field () to be the intersection of all σ–fields, which contain all elements of . Obviously, () is an algebra.        QED

A collection of subsets of is called a monotonic class if the limit of any monotone sequence in belongs to .

If is a collection of subsets of , let * () denote the smallest monotonic class containing .

Lemma 1.2.2. A necessary and sufficient condition of an algebra to be a σ–field is that it is a monotonic class.

Proof.   (i) Obviously, if is a σ–field, it is a monotonic class.

(ii) Let be a monotonic class.

Let En , n ≥ 1. Define . Obviously Bn Bn+1 for all n ≥ 1. Hence . But . Thus, , En . Similarly, En . Thus, is a σ–field.        QED

Theorem 1.2.1. Let be an algebra. Then * () = (), where () is the smallest σ–field containing .

Proof.   See Shiryayev (1984, p. 139).

The measurable space (, ), where is the real line and = (), called the Borel measurable space, plays a most important role in the theory of statistics. Another important measurable space is (n, n), n ≥ 2, where n = × × ··· × is the Euclidean n–space, and n = × ··· × is the smallest σ–field containing n, , and all n–dimensional rectangles I = I1 × ··· × In, where

The measurable space (∞, ∞) is used as a basis for probability models of experiments with infinitely many trials. ∞ is the space of ordered sequences x = (x1, x2, …), −∞ < xn < ∞, n = 1, 2, …. Consider the cylinder sets

and

where Bi are Borel sets, i.e., Bi . The smallest σ–field containing all these cylinder sets, n ≥ 1, is (∞). Examples of Borel sets in (∞) are

(a) {x: x ∞, , xn > a}

or

(b) {x: x ∞, , xn ≤ a}.

1.3 PROBABILITY SPACES

Given a measurable space (, ), a probability model ascribes a countably additive function P on , which assigns a probability P{A} to all sets A . This function should satisfy the following properties.

(1.3.1)

(1.3.2)

Recall that if A B then P {A} ≤ P{B}, and P{} = 1 − P{A}. Other properties will be given in the examples and problems. In the sequel we often write AB for A B.

Theorem 1.3.1. Let (, , P) be a probability space, where is a σ–field of subsets of and P a probability function....

Produkt-Details

• Erscheint lt. Verlag: 17.12.2013
• Reihe/Serie: Wiley Series in Probability and Statistics
• Reihe/Serie: Wiley Series in Probability and Statistics
• Reihe/Serie: Wiley Series in Probability and Statistics
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: Angewandte Wahrscheinlichkeitsrechnung u. Statistik • Applied Probability & Statistics • Approach • authority • clear • Electrical & Electronics Engineering • Elektrotechnik u. Elektronik • established • Exercises • fundamentals • GAP • important • Kriminologie • Mathematical • Necessary • Numerical Methods & Algorithms • Numerische Methoden u. Algorithmen • numerous problemsolving • Probability • Probability & Mathematical Statistics • Problems • Skills • Statistical • Statistics • Statistik • theory • topic • uniquely • Wahrscheinlichkeitsrechnung u. mathematische Statistik
• ISBN-13: 9781118605837 / 9781118605837