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2024 | OriginalPaper | Chapter

1. Preliminaries

Authors : Iickho Song, So Ryoung Park, Wenyi Zhang, Seungwon Lee

Published in: Fundamentals of Order and Rank Statistics

Publisher: Springer Nature Switzerland

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Abstract

In this chapter, we address and review key concepts that will be used in later chapters. In Sect. 1.1, we review briefly the key notions of probability functions and moments of random vectors. Then, we introduce step and impulse functions, which will prove useful in describing and analyzing non-linear statistics, in Sect. 1.2. Section 1.3 is devoted to the discussion on signs and magnitudes of random variables, which will play an important role in describing and understanding order and rank statistics. In the discussion, we will find interesting applications of the step and impulse functions. In Sect. 1.4, we introduce concisely the notions of order statistics, magnitude order statistics, ranks, and magnitude ranks of random vectors, which are the themes of the following chapters.

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The terminology ‘probability function’, ‘distribution function’, and ‘probability measure’ are often used interchangeably. In this book, ‘probability function’ indicates the cdf, pdf, pdmf, or pmf; ‘distribution function’ is another name of the cdf; and ‘probability measure’ denotes the set function \(\mathsf {P} (\cdot )\).

Here, fair means ‘the two sample points head and tail are equally likely to occur’.

Two sets are called mutually exclusive, or disjoint, if they have no common element.

The class of all the subsets of a set is called the power set of the set. The power set of a set A is often denoted by \(2^{A}\).

Because the probability measure \(\mathsf {P}\) is a set function, \(\mathsf {P} (\{k\})\) and \(\mathsf {P} ( \{\mathit {head} \} )\), for instance, are the exact expressions. Nonetheless, the expressions \(\mathsf {P} (k)\), \( \mathsf {P} \{k\}\), \(\mathsf {P} ( \mathit {head} )\), and \(\mathsf {P} \{ \mathit {head} \}\) are also used.

The sigma algebra that contains all open intervals in \(\mathbb {R}\) is called the Borel algebra, Borel sigma field, or Borel field of \(\mathbb {R}\).

When there is more than one subscript, commas are needed in some cases: for example, the joint pdf \(f_{X,Y}\) of \((X, Y)\) should be differentiated from the pdf \(f_{XY}\) of the product XY . In other cases, commas are unnecessary: for instance, \(R_{XY}\), \(\mu _{jk}\), \(K_{XY}\), \(\cdots \) denote relations among two or more random variables, but is expressed without commas.

When we take complex functions also into account, the notation \(\left < a(x), b(x) \right >\) is defined as \(\left < a(x), b(x) \right > = \int _{-\infty }^{\infty } a(x) b^{*}(x) dx\).

In this book, we reserve the term distribution for another concept in probability.

In the narrower sense, when we say ‘joint’ pdf \(f_{X,Y}\), it is usually assumed that X and Y are distinct random variables. In the joint pdf (1.2.58), we adopt the term ‘joint’ in a wider sense.

More generally, \(\int _{0}^{\infty } x^m f_X(x) dx\) for \(m \in \mathbb {J}_{1,\infty }\) are called the half moments, incomplete moments, or partial moments.

More generally, the gamma function can be defined as \(\varGamma (\alpha ) = \lim \limits _{n \to \infty } \frac { n^{\alpha -1} n!} {(\alpha )_{n}} = \lim \limits _{n \to \infty } \frac { n^{\alpha } n!} {(\alpha )_{n+1}} \) for a complex number \(\alpha \) such that \(\alpha \notin \mathbb {J}_{-\infty , 0}\), where \((\alpha )_n\) denotes the rising factorial defined by \((\alpha )_n =1\) for \(n=0\) and \((\alpha )_n = \alpha (\alpha +1) \cdots (\alpha +n-1)\) for \(n \in \mathbb {J}_{1,\infty }\).

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R.N. McDonough, A.D. Whalen, Detection of Signals in Noise, 2nd edn. (Academic, New York, 1995)

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Title: Preliminaries
Authors: Iickho Song
So Ryoung Park
Wenyi Zhang
Seungwon Lee
Publisher: Springer Nature Switzerland
Book: Fundamentals of Order and Rank Statistics
Print ISBN: 978-3-031-50600-0

Electronic ISBN: 978-3-031-50601-7

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-3-031-50601-7_1

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