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→Log-concave distributions: Use ≥ symbol rather than ">=". Add URLs and other details to references. |
→Log-concave distributions: I add a supplement that binomial distribution is also a log concave distribution |
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==Properties==
* A log-concave function is also [[Quasi-concave function|quasi-concave]]. This follows from the fact that the logarithm is monotone implying that the [[Level set#Sublevel and superlevel sets|superlevel sets]] of this function are convex.<ref name=":0" />
* 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(−''x''<sup>2</sup>/2)}} which is log-concave since {{math|log ''f''(''x'')}} = {{math|−''x''<sup>2</sup>/2}} is a concave function of {{math|''x''}}. But {{math|''f''}} is not concave since the [[second derivative]] is positive for |{{math|''x''}}| > 1:
::<math>f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0</math>
* From above two points, [[Concave function|concavity]] <math>\Rightarrow</math> log-concavity <math>\Rightarrow</math> [[Quasiconcave function|quasiconcavity]].
* 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 |first1=Stephen |last1=Boyd |author-link=Stephen P. Boyd |first2=Lieven |last2=Vandenberghe |chapter=Log-concave and log-convex functions |title=Convex Optimization |publisher=Cambridge University Press |year=2004 |isbn=0-521-83378-7 |chapter-url=https://web.stanford.edu/~boyd/cvxbook/ |pages=104–108 }}</ref>
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*the [[exponential distribution]],
*the [[uniform distribution (continuous)|uniform distribution]] over any [[convex set]],
*the [[binomial distribution]],
*the [[logistic distribution]],
*the [[extreme value distribution]],
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*the [[Wishart distribution]], if ''n'' ≥ ''p'' + 1,<ref name="prekopa">{{cite journal | last1 = Prékopa | first1 = András | author-link = András Prékopa | year = 1971 | title = Logarithmic concave measures with application to stochastic programming | journal = [[Acta Scientiarum Mathematicarum]] | volume = 32 | issue = 3-4 | pages = 301–316 | url = http://rutcor.rutgers.edu/~prekopa/SCIENT1.pdf}}</ref>
*the [[Dirichlet distribution]], if all parameters are ≥ 1,<ref name="prekopa"/>
*the [[gamma distribution]] if the [[shape parameter]] is ≥ 1,
*the [[chi-square distribution]] if the number of degrees of freedom is ≥ 2,
*the [[beta distribution]] if both shape parameters are ≥ 1, and
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