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

Abstract

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.