Logarithmically concave function: Difference between revisions

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==Properties==

* A positive log-concave function is also [[Quasi-concave_functionconcave function| quasi-concave]].

* Every concave function that is nonnegative on its ___domain is log-concave. However, the reverse does not necessarily hold. An example is the [[Gaussian function]] {{math|''f''(''x'')}} = {{math|exp(−x²/2)}} which is log-concave since {{math|log ''f''(''x'')}} = {{math|−''x''²/2}} is a concave function of {{math|''x''}}. But {{math|''f''}} is not concave since the second derivative is positive for |{{math|''x''}}| > 1:

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::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>

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

::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>, <ref> Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF) p.105</ref>

:i.e.

::<math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is

:[[positive-definite matrix|negative semi-definite]]. For functions of one variable, this condition simplifies to

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* Products: The product of log-concave functions is also log-concave. Indeed, if {{math|''f''}} and {{math|''g''}} are log-concave functions, then {{math|log&nbsp;''f''}} and {{math|log&nbsp;''g''}} are concave by definition. Therefore

::<math>\log\,f(x) + \log\,g(x) = \log(f(x)g(x))</math>

:is concave, and hence also {{math|''f''&nbsp;''g''}} is log-concave.

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|last1=Dharmadhikari|first1=Sudhakar

|last2=Joag-Dev

|first2=Kumar|series=Probability and Mathematical Statistics

|series=Probability and Mathematical Statistics

|publisher=Academic Press, Inc.

|___location=Boston, MA

|year=1988

|pages=xiv+278

|isbn=0-12-214690-5|mr=954608}}

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

|authorlink= <!-- Johann Pfanzagl -->

|last2=with the assistance of R. Hamböker

|year=1994|publisher=Walter de Gruyter

|publisher=Walter de Gruyter

|isbn=3-11-013863-8

|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=Frank Proschan -->

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

|volume=187

|publisher=Academic Press, Inc.

|___location=Boston, MA