Staudte Robert G, Xia Aihua
Department of Mathematics and Statistics, La Trobe University, Bundoora, VIC 3086, Australia.
School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
Entropy (Basel). 2018 Apr 25;20(5):317. doi: 10.3390/e20050317.
We demonstrate that questions of convergence and divergence regarding shapes of distributions can be carried out in a location- and scale-free environment. This environment is the class of probability density quantiles (pdQs), obtained by normalizing the composition of the density with the associated quantile function. It has earlier been shown that the pdQ is representative of a location-scale family and carries essential information regarding shape and tail behavior of the family. The class of pdQs are densities of continuous distributions with common domain, the unit interval, facilitating metric and semi-metric comparisons. The Kullback-Leibler divergences from uniformity of these pdQs are mapped to illustrate their relative positions with respect to uniformity. To gain more insight into the information that is conserved under the pdQ mapping, we repeatedly apply the pdQ mapping and find that further applications of it are quite generally entropy increasing so convergence to the uniform distribution is investigated. New fixed point theorems are established with elementary probabilistic arguments and illustrated by examples.
我们证明，关于分布形状的收敛和发散问题可以在无位置和无尺度的环境中进行。这种环境是概率密度分位数（pdQ）的类别，它通过用相关的分位数函数对密度的组成进行归一化而获得。 earlier已表明，pdQ代表一个位置-尺度族，并携带有关该族形状和尾部行为的基本信息。pdQ的类别是具有共同定义域（单位区间）的连续分布的密度，便于进行度量和半度量比较。这些pdQ与均匀性的Kullback-Leibler散度被映射以说明它们相对于均匀性的相对位置。为了更深入地了解在pdQ映射下守恒的信息，我们反复应用pdQ映射，并发现其进一步应用通常会增加熵，因此研究了向均匀分布的收敛。用基本的概率论证建立了新的不动点定理，并通过例子进行了说明。
