Geometrically concave univariate distributions

Research output: Contribution to journalArticle

27 Citations (Scopus)

Abstract

In this paper our aim is to show that if a probability density function is geometrically concave (convex), then the corresponding cumulative distribution function and the survival function are geometrically concave (convex) too, under some assumptions. The proofs are based on the so-called monotone form of l'Hospital's rule and permit us to extend our results to the case of the concavity (convexity) with respect to Hölder means. To illustrate the applications of the main results, we discuss in details the geometrical concavity of the probability density function, cumulative distribution function and survival function of some common continuous univariate distributions. Moreover, at the end of the paper, we present a simple alternative proof to Schweizer's problem related to the Mulholland's generalization of Minkowski's inequality.

Original languageEnglish
Pages (from-to)182-196
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume363
Issue number1
DOIs
Publication statusPublished - Mar 1 2010

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Survival Function
Concavity
Cumulative distribution function
Probability density function
Distribution functions
Univariate
Minkowski's inequality
Convexity
Monotone
Alternatives
Generalization
Form

Keywords

  • Convexity (concavity) with respect to Hölder means
  • Geometrically concave (convex) functions
  • Log-concave (log-convex) functions
  • Monotone form of l'Hospital's rule
  • Mulholland's inequality
  • Statistical distributions

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

Geometrically concave univariate distributions. / Baricz, A.

In: Journal of Mathematical Analysis and Applications, Vol. 363, No. 1, 01.03.2010, p. 182-196.

Research output: Contribution to journalArticle

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