ON THE NORMALITY OF TRANSFORMED BETA AND UNIT-GAMMA RANDOM-VARIABLES

In this paper we consider properties of the logarithmic and Tukey's lambda-type transformations of random variables that follow beta or unit-gamma distributions. Beta distributions often arise as models for random proportions, and unit-gamma distributions, although not well-known, may serve the same purpose. The latter possess many properties similar to those of beta distributions. Some transformations of random variables that follow a beta distribution are considered by Johnson (1949) and Johnson and Kotz (1970, 1973). These are used to obtain a "new" random variable that potentially approximately follows a normal distribution, so that practical analyses become possible. We study normality-related properties of the above transormations. This is done for the first time for unit-gamma distributions. Under the logarithmic transformation the beta and unit-gamma distributions become, respectively, the logarithmic F and generalized logistic distributions. The distributions of the transformed beta and unit-gamma distributions after application of Tukey's lambda-type transformations cannot be derived easily; however, we obtain the first four moments and expressions for the skewness and kurtosis of the transformed variables. Values of skewness and kurtosis for a variety of different parameter values are calculated, and in consequence, the near (or not near) normality of the transformed variables is evaluated. Comments on the use of the various transformations are provided.