Skewness in commingled distributions

Abstract

A likelihood ratio test is given for distinguishing skewness from commingled distributions, using a power transform to remove skewness appropriately for each of the alternatives tested. The alternative hypotheses postulate that the transformed data are from one normal or a mixture of two or three normal homoscedastic distributions. Since each mixture has unique asymmetry, skewness is estimated simultaneously with the means, proportions and variance of components. Commingling cannot be rigorously proven in this way, as some other transform may provide a better approximation to normality. However, the error of asserting admixture whenever there is skewness has been avoided, and estimates of admixture parameters provide a basis for more conclusive tests in relatives or other populations. Two examples are given, one in which adjustment for skeweness left evidence of commingling.