The shapes of things to come: probability density quantiles

Abstract

For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQ s) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback-Leibler divergences. Convergent empirical estimates of these pdQ s are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQ s from the symmetric class. Further, a precise classification of shapes by tail behavior can be defined simply in terms of pdQ boundary derivatives.

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

“…When the density function f is bounded, we showed that each application lowers its modal height and hence the resulting density function is closer to the uniform density than f . Furthermore, we established a necessary and sufficient condition for converging in norm to the uniform density, giving a positive answer to a conjecture raised in [ 1 ]. In particular, if f is bounded, we proved that converges in norm to the uniform density for any .…”

“…It is known that the Tukey( ) distributions, with , are good approximations to Student’s t -distributions for provided is chosen properly. The same is true for their corresponding pdQs ([ 1 ], Section 3.2). For example, the pdQof with degrees of freedom is well approximated by the choice Its location is marked by the small black disk in Figure 1 ; it is of distance 2 from uniformity.…”

“…variables, each from a distribution with density f , it is feasible to carry out a nonparametric test of uniformity by estimating the pdQ with a kernel density estimator and comparing it with the uniform density on [0,1] using any one of a number of metrics or semi-metrics. Consistent estimators for based on normalized reciprocals of the quantile density estimators derived in [ 11 ] are available and described in (Staudte [ 1 ], Section 2). Note that such a test compares an arbitrary uniform distribution with an arbitrary member of the location-scale family generated by f ; it is a test of shape only.…”

“…In addition, the tail behaviour of the original family can be described in terms of the boundary derivatives of its pdQ. Empirical estimators of the pdQs enable one to carry out inference, such as robust fitting of shape parameter families to data; details are in [ 1 ].…”

An efficient estimator of the parameters of the Generalized Lambda Distribution

“…Dedduwakumara et al (2019) introduced a method for choosing optimal parameters for generalized distributions to approximate other distributions. The method uses the probability density quantile function (pdQ, Staudte, 2017) to first find the optimal shape parameters and is computationally quick, simple to implement and often outperforms other methods. Motivated by this, we introduce estimators of the GLD shape parameters arising from the estimated pdQ.…”