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Symmetrizing Positive Random Variables

Transformations to symmetry are frequently used in statistics to improve the accuracy of statistical models that assume the normal distribution or at least symmetrically distributed errors. Typically the transformation is chosen after the data has been observed but it is of theoretical interest to see what is the best symmetrizing transformation for various possible distributions.
We show the probability density function for the transformed random variable, ,
,
where and is a random variable with support on . This table shows the value of for each of the distributions that makes skewness of close to zero for the distributions used.
For the first three distributions, the usual skewness coefficient is defined for , so the best to symmetrize the distribution was found using Mathematica's built-in function FindMinimum to minimize the absolute skewness coefficient. The skewness coefficient may be computed using Mathematica's built-in function NIntegrate to obtain the first three moments. In the case of the inverse gamma distribution the mean and variance are undefined so a different definition of skewness is needed. The simplest definition for this problem is to take the difference in areas with respect to the mode. Using this definition, was obtained. The values given in the table agree visually with the display.
In general, the value of to symmetrize the distribution may depend on the shape parameter. Of course, for some distributions, such as bimodal ones, there may not exist a symmetrizing transformation of the type discussed in this Demonstration.
It is of interest to note that increasing pushes the right tail of the distribution out toward and pulls the left tail in towards 0. Decreasing reverses this effect so the right tail is pulled in and the left tail is pushed out toward . This effect is sometimes referred to as the ladder of transformations with increasing/decreasing corresponding to going up/down the ladder.

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Let be the probability density function of with support on and let be defined by
.
Then for , the probability density function for may be written
with support on for and on for . For , the probability density function is with support on . The derivation of this result is within the scope of a first course in mathematical statistics.
Power transformations, ,
are frequently used to improve the statistical assumption of normal or symmetrically distributed response variables in many situations in applied statistics. The family of Box–Cox transformations [1] given by
is often preferred for mathematical analysis and visualization since they are monotonic (or order preserving) and continuous at . But for interpretability, it is sometimes more convenient to use a simple member of the power transformation family, for example, square root or log.
It is interesting that for the four distributions considered, the variance is proportional to or approximately proportional to the square of the mean and hence the logarithmic transformation is the variance-stabilizing transformation [2]. This Demonstration shows that the best symmetrizing transformation is not so simple and, further, lets you compare these transformations graphically.
[1] G. E. P. Box and D. R. Cox, "An Analysis of Transformations," Journal of the Royal Statistical Society B, 26(2), 1964 pp. 211–252.
[2] M. S. Bartlett, "The Use of Transformations," Biometrics, 3(1), 1947 pp. 37–52.
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