Power transformations: An application for symmetrizing the distribution of sample coefficient of variation from inverse gaussian populations

The coefficient of variation (CV) of a random variable (or that of the corresponding
population), defined to be the ratio of the standard deviation to the mean of the cor­
responding population, has been used in wide­ranging applications in many areas
of applied research including agro­biological, industrial, social, and economic
research (Johnson et  al. 1994, Chapter 15). In these applications, the random vari­
able of interest is assumed to follow a Gaussian distribution that is symmetric and
has support on the whole real number line (see Laubscher 1960, Singh 1993, Johnson
et al. 1994, Chaubey et al. 2013). However, in many of these applications, the random
variable may be more appropriately modeled by a distribution, which is positively
skewed and is supported on the positive half. To model such situations, use of an
inverse Gaussian (IG) distribution is often more justified compared to lognormal,
gamma, and Weibull distributions (see Chhikara and Folks 1977, 1989, Kumagai
et  al. 1996, Takagi et  al. 1997).
The purpose of this chapter is to review the properties of variance stabilizing
and skewness­reducing transformations for CV in the context of the IG population
as investigated recently by Chaubey et  al. (2014b). The variables observed for
evaluation of genetic resources and modeling climate data often need transformation
so that the associated assumptions in applying the statistical methods are tenable.