Fundamentals of Order and Rank Statistics

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

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is devoted to the fundamentals of order and rank statistics. Primarily focusing on theoretical properties, it also discusses practical aspects, including interesting applications of step and impulse functions for the distribution of random variables, magnitudes, and signs. New concepts are introduced, such as independent and semi-identically distributed random vectors and probability density-mass functions. This book also presents an investigation of relative magnitudes in order statistics, and of correlation coefficients among signs, ranks, and magnitudes. The basic concepts are described in clear terms, and step-by-step details are provided for most of the presented mathematical results. The exposition is accompanied by numerous examples and more than 100 exercises, for which a complete solution manual is available. Providing a useful reference, and requiring only a basic understanding of probability and random variables, the book will appeal to a wide readership.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

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.

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

Chapter 2. Order Statistics

Abstract

In this chapter, recollecting the notion of order statistics introduced in Sect. 1.4, we discuss the distributions and properties of order statistics in detail. As mentioned in Sect. 1.4, in dealing with the order statistics, we assume that the dimension, or size, of the random vectors in this chapter is \(n \in \mathbb {J}_{2, \infty }\) unless specified otherwise. When it becomes appropriate, we denote the size n of the random vector more explicitly.

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

Chapter 3. Rank Statistics

Abstract

Ranks and magnitude ranks are closely related with order and magnitude order statistics discussed in Chap. 2, and are often used together with sign statistics. In this chapter, we first address the distributions of ranks and magnitude ranks in Sect. 3.1. Then, the notion of score functions are considered in Sect. 3.2, focusing mostly on locally optimum score functions. In Sect. 3.3, we consider the correlation coefficients among various statistics from i.i.d. random vectors.

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

Backmatter

Titel: Fundamentals of Order and Rank Statistics
verfasst von: Iickho Song
So Ryoung Park
Wenyi Zhang
Seungwon Lee
Verlag: Springer Nature Switzerland
Electronic ISBN: 978-3-031-50601-7
Print ISBN: 978-3-031-50600-0
DOI: https://doi.org/10.1007/978-3-031-50601-7

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Chapter 2. Order Statistics

Chapter 3. Rank Statistics

Backmatter

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