Effect of power transformations on a Weibull-distributed error component of a multiplicative error model

The error component of a Multiplicative Error Model (MEM) can possibly be a Weibull distribution ( ;   and   are shape and scale parameters respectively). Data transformation is a popular remedial measure to stabilize the variance of a data set prior to statistical modeling. Therefore, in this paper the effects of power transformations on the mean and variance of a Weibull distributed error component of a MEM are investigated. The popular transformations - inverse, square-root, inverse-square-root, square, inverse-square, cube root, inverse cube root, cube and inverse cube transformations were studied. The probability density function (pdf) and the kth raw moment of the p-th power–transformed Weibull random variable are obtained. The mean and variance of     and those of the power – transformed distributions are calculated for  = 6, 7, . ., 99, 100 with the corresponding values of n for which the mean of the untransformed distribution is equal to one. The relative changes in mean and variance are used for the investigations. For all the transformations, the means of the power transformed distributions are not different from 1. For variances, it was found that there are relative increases for the inverse, square, inverse square, cube and inverse cube transformations. However, the square-root, inverse square root, cube root and inverse-cube-root transformations decreased the variance relative to the variance of the untransformed distribution. This paper concludes that the square-root, inverse square root, cube root and inverse-cube-root transformations would yield better results as they reveal constancy in variance when using MEM with a Weibull distributed error component and where data transformation is deemed necessary to stabilize the variance of the data set.

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