# Asymptotics of the Sample Coefficient of Variation and the Sample Dispersion

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State: Serval

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State: Serval

Version: author

Serval ID

serval:BIB_A5E97B82B245

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

Asymptotics of the Sample Coefficient of Variation and the Sample Dispersion

Journal

Journal of Statistical Planning and Inference

ISSN

0378-3758

Publication state

Published

Issued date

2010

Peer-reviewed

Oui

Volume

140

Number

2

Pages

358-368

Language

english

Abstract

The coefficient of variation and the dispersion are two examples of widely used measures of variation. We show that their applicability in practice heavily depends on the existence of sufficiently many moments of the underlying distribution. In particular, we offer a set of results that illustrate the behavior of these measures of variation when such a moment condition is not satisfied. Our analysis is based on an auxiliary statistic that is interesting in its own right. Let (X-i)(i >= 1) be a sequence of positive independent and identically distributed random variables with distribution function F and define for n is an element of N

Tn := X-1(2) + X-2(2) + ... + X-n(2)/(X-1 + X-2 + ... + X-n)(2).

Mainly using the theory of functions of regular variation, we derive weak limit theorems for the properly normalized random quantity T-n. given that 1 - F is regularly varying. Following a distributional approach based on T-n, we then analyze asymptotic properties of the sample coefficient of variation. As a second illustration of the same method, we then turn to the sample dispersion. We also include asymptotic properties of the first moments of these quantities. Finally, we give a distributional result on Student's t-statistic which is closely related to T-n. The main message of this paper is to show that the unconscientious use of some measures of variation can lead to wrong conclusions.

Tn := X-1(2) + X-2(2) + ... + X-n(2)/(X-1 + X-2 + ... + X-n)(2).

Mainly using the theory of functions of regular variation, we derive weak limit theorems for the properly normalized random quantity T-n. given that 1 - F is regularly varying. Following a distributional approach based on T-n, we then analyze asymptotic properties of the sample coefficient of variation. As a second illustration of the same method, we then turn to the sample dispersion. We also include asymptotic properties of the first moments of these quantities. Finally, we give a distributional result on Student's t-statistic which is closely related to T-n. The main message of this paper is to show that the unconscientious use of some measures of variation can lead to wrong conclusions.

Keywords

Weak limit theorems, Functions of regular variation, Domain of attraction of a stable law, Sample coefficient of variation, Sample dispersion, Student's t-statistic, Extreme value theory

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Create date

31/08/2009 12:39

Last modification date

03/03/2018 19:16