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

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Serval ID
serval:BIB_A5E97B82B245
Type
Article: article from journal or magazin.
Collection
Publications
Institution
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.
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
Web of science
Open Access
Yes
Create date
31/08/2009 12:39
Last modification date
20/08/2019 15:11