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

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 associate...

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Autores principales: Staudte, Robert G., Xia, Aihua
Formato: Online Artículo Texto
Lenguaje:English
Publicado: MDPI 2018
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512836/
https://www.ncbi.nlm.nih.gov/pubmed/33265408
http://dx.doi.org/10.3390/e20050317
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author Staudte, Robert G.
Xia, Aihua
author_facet Staudte, Robert G.
Xia, Aihua
author_sort Staudte, Robert G.
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description 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.
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spelling pubmed-75128362020-11-09 Divergence from, and Convergence to, Uniformity of Probability Density Quantiles Staudte, Robert G. Xia, Aihua Entropy (Basel) Article 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. MDPI 2018-04-25 /pmc/articles/PMC7512836/ /pubmed/33265408 http://dx.doi.org/10.3390/e20050317 Text en © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
spellingShingle Article
Staudte, Robert G.
Xia, Aihua
Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title_full Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title_fullStr Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title_full_unstemmed Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title_short Divergence from, and Convergence to, Uniformity of Probability Density Quantiles
title_sort divergence from, and convergence to, uniformity of probability density quantiles
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512836/
https://www.ncbi.nlm.nih.gov/pubmed/33265408
http://dx.doi.org/10.3390/e20050317
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