The impact of power transformations on the parameters of the gamma distributed error component of a multiplicative error model

Abstract

Considering that the error component of a Multiplicative Error Model(MEM) can possibly be
a gamma distribution (G(α, β); α and β are shape and scale parameters respectively). This paper studies
the effects of power transformations on the mean and variance of a gamma distributed error component.
The popular transformations: square-root, Wilson-Hilferty, inverse-square-root, inverse, inverse-square and
square transformations were studied. The probability density function(pdf) and the kth-uncorrected mo
ment of the p-th power –transformed gamma random variable are obtained. The mean and variance of
G(α, β) and those of the power – transformed distributions are calculated for α = 5, 6, · · · , 99, 100 with
the corresponding values of β for which the mean of the untransformed distribution is equal to one. The
effects of the power transformations on the mean and variance of the gamma distribution are investigated
for α ≥ 9 where all the transformed distributions have a unit mean value. The relative changes in mean
and variance are used for the investigations. For all the transformations, there are no changes in the mean.
For variances: it was found that there are relative increases for the inverse, inverse square and square
transformations. However, the square-root, Wilson Hilferty and inverse-square-root transformations de
creased the variance relative to the variance of the untransformed distribution. This paper concludes that
the square-root, Wilson Hilferty and inverse-square-root transformations would yield better results when
using MEM that the error component assumes a gamma distribution and where the goal is to stabilize the
variance of the data set through data transformation.