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2024 | OriginalPaper | Buchkapitel

1. Preliminaries

verfasst von : Iickho Song, So Ryoung Park, Wenyi Zhang, Seungwon Lee

Erschienen in: Fundamentals of Order and Rank Statistics

Verlag: 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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Nächstes Kapitel Order Statistics

Nur mit Berechtigung zugänglich

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 \(n \in \mathbb {J}_{1,\infty }\).

M. Abramowitz, I.A. Stegun (ed.), Handbook of Mathematical Functions (Dover, New York, 1972)

N. Balakrishnan, Handbook of the Logistic Distribution (Marcel Dekker, New York, 1992)

J.L. Challifour, Generalized Functions and Fourier Analysis: An Introduction (W. A. Benjamin, Reading, 1972)

H.A. David, Order Statistics, 2nd edn. (John Wiley and Sons, New York, 1980)

H.A. David, H.N. Nagaraja, Order Statistics, 3rd edn. (John Wiley and Sons, New York, 2003)CrossRef

W.A. Gardner, Introduction to Random Processes with Applications to Signals and Systems, 2nd edn. (McGraw-Hill, New York, 1990)

I.M. Gelfand, I. Moiseevich, Generalized Functions (Academic, New York, 1964)

R.M. Gray, L.D. Davisson, An Introduction to Statistical Signal Processing (Cambridge University Press, Cambridge, 2010)

H.J. Godwin, Some low moments of order statistics. Ann. Math. Stat. 20(2), 279–285 (1949)MathSciNetCrossRef

10.

J. Hajek, Nonparametric Statistics (Holden-Day, San Francisco, 1969)

11.

J. Hajek, Z. Sidak, P.K. Sen, Theory of Rank Tests, 2nd edn. (Academic, New York, 1999)

12.

T.P. Hettmansperger, Statistical Inference Based on Ranks (John Wiley and Sons, New York, 1984)

13.

R.F. Hoskins, J.S. Pinto, Theories of Generalised Functions (Horwood, Chichester, 2005)

14.

D.S. Jones, The Theory of Generalised Functions, 2nd edn. (Cambridge University Press, Cambridge, 1982)CrossRef

15.

R.P. Kanwal, Generalized Functions: Theory and Applications (Birkhauser, Boston, 2004)CrossRef

16.

A.B. Kharazishvili, Nonmeasurable Sets and Functions (Elsevier, Amsterdam, 2004)

17.

M.J. Lighthill, An Introduction to Fourier Analysis and Generalised Functions (Cambridge University Press, Cambridge, 1980)

18.

E. Lukacs, Characteristic Functions, 2nd edn. (Griffin, London, 1970)

19.

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

20.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962)

21.

I. Song, S.R. Park, S. Yoon, Probability and Random Variables: Theory and Applications (Springer-Verlag, Berlin, 2022)CrossRef

22.

G.L. Wies, E.B. Hall, Counterexamples in Probability and Real Analysis (Oxford University Press, New York, 1993)CrossRef

Titel: Preliminaries
verfasst von: Iickho Song
So Ryoung Park
Wenyi Zhang
Seungwon Lee
Verlag: Springer Nature Switzerland
Buch: Fundamentals of Order and Rank Statistics
Print ISBN: 978-3-031-50600-0

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

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

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