Logarithmically concave function: Difference between revisions

Revision as of 22:01, 30 January 2020

Revision as of 20:29, 10 December 2020

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* A twice differentiable, nonnegative function with a convex ___domain is log-concave if and only if for all {{math|''x''}} satisfying {{math|''f''(''x'') > 0}},

::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>,<ref name=":0">{{cite book |first=Stephen |last=Boyd |authorlinkauthor-link=Stephen P. Boyd |first2=Lieven |last2=Vandenberghe |chapter=Log-concave and log-convex functions |title=Convex Optimization |___location= |publisher=Cambridge University Press |year=2004 |isbn=0-521-83378-7 |chapter-url=https://web.stanford.edu/~boyd/cvxbook/ |pages=104–108 }}</ref>

:i.e.

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*The [[chi distribution]].

*The [[hyperbolic secant distribution]].

*The [[Wishart distribution]], where ''n'' >= ''p'' + 1.<ref name="prekopa">{{cite journal | last1 = Prékopa | first1 = András | year = 1971 | title = Logarithmic concave measures with application to stochastic programming | url = | journal = Acta Scientiarum Mathematicarum | volume = 32 | issue = | pages = 301–316 }}</ref>

*The [[Dirichlet distribution]], where all parameters are >= 1.<ref name="prekopa"/>

*The [[gamma distribution]] if the shape parameter is >= 1.

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==References==

* {{cite book|authorlinkauthor-link=Ole Barndorff-Nielsen|last=Barndorff-Nielsen|first=Ole|title=Information and exponential families in statistical theory|series=Wiley Series in Probability and Mathematical Statistics|publisher=John Wiley \& Sons, Ltd.|___location=Chichester|year=1978|pages=ix+238 pp|isbn=0-471-99545-2|mr=489333}}

* {{cite book|title=Unimodality, convexity, and applications

|last1=Dharmadhikari|first1=Sudhakar

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* {{cite book|title=Parametric Statistical Theory | last1=Pfanzagl | first1=Johann

|authorlinkauthor-link= <!-- Johann Pfanzagl -->

|last2=with the assistance of R. Hamböker

|year=1994|publisher=Walter de Gruyter

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|mr=1291393}}

* {{cite book|title=Convex functions, partial orderings, and statistical applications|last1=Pečarić|first1=Josip E.|last2=Proschan|first2=Frank|last3=Tong|first3=Y. L.|<!-- authorlink2|author-link2=Frank Proschan -->

|series=Mathematics in Science and Engineering|volume=187

|publisher=Academic Press, Inc.