The impact of power transformations on the parameters of the gamma distributed error component of a multiplicative error model - Ohakwe, J.; Ajibade, F. B. (Benin Journal of Statistics, 2018)
Considering that the error component of a Multiplicative Error Model(MEM) can possibly bea gamma distribution (G(α, β); α and β are shape and scale parameters respectively). This paper studiesthe 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 andsquare transformations were studied. The probability density function(pdf) and the kth-uncorrected moment of the p-th power –transformed gamma random variable are obtained. The mean and variance ofG(α, β) and those of the power – transformed distributions are calculated for α = 5, 6, · · · , 99, 100 withthe corresponding values of β for which the mean of the untransformed distribution is equal to one. Theeffects of the power transformations on the mean and variance of the gamma distribution are investigatedfor α ≥ 9 where all the transformed distributions have a unit mean value. The relative changes in meanand 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 squaretransformations. However, the square-root, Wilson Hilferty and inverse-square-root transformations decreased the variance relative to the variance of the untransformed distribution. This paper concludes thatthe square-root, Wilson Hilferty and inverse-square-root transformations would yield better results whenusing MEM that the error component assumes a gamma distribution and where the goal is to stabilize thevariance of the data set through data transformation.
Keywords: multiplicative error model; gamma distribution; power transformation; mean; variance