Unit Mean and Constant Variance of the Generalized Gamma Distribution after Square Root Transformation in Statistical modeling

Abstract

In this paper, we studied the effect of square root transformation on the error component of the multiplicative error model whose distribution belongs to the generalized gamma family. The purpose of the study is to determine the effect of the said transformation on the basic assumptions; unit mean and constant variance required for statistical modeling. The special cases of the Generalized Gamma Distribution considered are the three-parameter Gamma distribution error component, the Chi-square, Exponential, Weibull, Rayleigh and Maxwell distributed error components. From the results of the study, the unit mean assumption is approximately maintained for all the distributions. It was also found that there were reduction in the variances of all the square root transformed distributions under study except those of the Gamma(a, b, 1), when a > 1, Rayleigh and Maxwell distributions that increased. Therefore we conclude that square root transformation is not appropriate for multiplicative error models with either a Gamma (a, b, 1), for a > 1 or Rayleigh or Maxwell distributed error component. Finally square root transformations where applicable are successful for the distributions under study if the variance of the transformed data < 0.5.

Keywords: Generalized gamma distribution; Square root transformation; Mean; Variance; multiplicative error Model; Error component