Asymptotics of the sample coefficient of variation and the sample dispersion

Abstract

MSC:primary 62G20 secondary 60F05 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 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 Ն 1 be a sequence of positive independent and identically distributed random variables with distribution function F and define for n ∈ N T n := X 2 1 + X 2 2 + · · · + X 2 n (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.

Taylor's law, via ratios, for some distributions with infinite mean

“…This approach is also used in de la Peña and Yang (1999). More closely related to our work, in the context of self-normalized processes, are Giné et al (1997), Fuchs and Joffe (1997), Albrecher and Teugels (2006), and Albrecher et al (2010). Proposition 1 is perhaps novel in pointing out the potential use of tilted distributions.…”

Long-term variation analysis of a tropical river’s annual streamflow regime over a 50-year period

“…SF2 represents the maximum annual flow throughout the sub-period; SF3 is the minimum annual flow over the sub-period; SF4 signifies the maximum monthly flow over a single year; SF5 corresponds to the minimum monthly flow over a single year; SD and CV are statistical measures of dispersion in a data series around its mean; and the CV of annual streamflow data series represents the ratio of standard deviation to the mean annual flow. The CV is valuable in matching the degree of dispersion and variation among data series (Albrecher et al 2010;Boik and Shirvani 2009). PR corresponds to the ratio between maximum and minimum streamflow values, and it provides initial indication of seasonal variability.…”

A combinatorial identity for a problem in asymptotic statistics

“…Moreover, T n is closely connected to the study of the sample coefficient of variation and the sample dispersion (cf. Albrecher, Ladoucette and Teugels [2]). …”