Divergence from, and Convergence to, Uniformity of Probability Density Quantiles.

The probability density quantile (pdQ) carries essential informationregarding shape and tail behavior of a location-scale family. Convergence ofrepeated applications of the pdQ mapping to the uniform distribution isinvestigated and new fixed point theorems are established. The Kullback-Leiblerdivergences from uniformity of these pdQs are mapped and found to beingredients in power functions of optimal tests for uniformity againstalternative shapes.