Titre
Asymptotics of the Sample Coefficient of Variation and the Sample Dispersion
Type
article
Institution
UNIL/CHUV/Unisanté + institutions partenaires
Auteur(s)
Albrecher, H.
Auteure/Auteur
Ladoucette, S.
Auteure/Auteur
Teugels, J.
Auteure/Auteur
Liens vers les personnes
Liens vers les unités
ISSN
0378-3758
Statut éditorial
Publié
Date de publication
2010
Volume
140
Numéro
2
Première page
358
Dernière page/numéro d’article
368
Peer-reviewed
Oui
Langue
anglais
Résumé
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.
PID Serval
serval:BIB_A5E97B82B245
Open Access
Oui
Date de création
2009-08-31T11:39:14.975Z
Date de création dans IRIS
2025-05-20T23:40:11Z
Fichier(s)![Vignette d'image]()
En cours de chargement...
Nom
BIB_A5E97B82B245.P001.pdf
Version du manuscrit
preprint
Taille
322.77 KB
Format
Adobe PDF
PID Serval
serval:BIB_A5E97B82B245.P001
URN
urn:nbn:ch:serval-BIB_A5E97B82B2455
Somme de contrôle
(MD5):adeac7c2d58209502f3a654c2422fc78