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→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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| last2=Molyboha | first2=Anton
| last3=Zabarankin | first3=Michael
| date=May 2009
| title=Maximum Entropy Principle with General Deviation Measures
| journal=[[Mathematics of Operations Research]]
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| issue=2
| pages=445–467
| doi=10.1287/moor.1090.0377
| url=https://www.researchgate.net/profile/Bogdan-Grechuk/publication/220442393_Maximum_Entropy_Principle_with_General_Deviation_Measures/links/59132b61a6fdcc963e7ed4fd/Maximum-Entropy-Principle-with-General-Deviation-Measures.pdf}}</ref>
As it happens, many common [[probability distribution]]s are log-concave. Some examples:<ref name=":1">See {{cite journal |first1=Mark |last1=Bagnoli |first2=Ted |last2=Bergstrom |year=2005 |title=Log-Concave Probability and Its Applications |journal=Economic Theory |volume=26 |issue=2 |pages=445–469 |doi=10.1007/s00199-004-0514-4 |s2cid=1046688 |url=http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf }}</ref>
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*the [[hyperbolic secant distribution]],
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*the [[Weibull distribution]] if the shape parameter is ≥ 1.
Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.
The following distributions are non-log-concave for all parameters:
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Note that the [[cumulative distribution function]] (CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:
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The following are among the properties of log-concave distributions:
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*If a multivariate density is log-concave, so is the [[marginal density]] over any subset of variables.
*The sum of two independent log-concave [[random variable]]s is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
*The product of two log-concave functions is log-concave. This means that [[joint distribution|joint]] densities formed by multiplying two probability densities (e.g. the [[normal-gamma distribution]], which always has a shape parameter
* If a density is log-concave, so is its [[survival function]].<ref name=":1" />
* If a density is log-concave, it has a monotone [[hazard rate]] (MHR), and is a [[Regular distribution (economics)|regular distribution]] since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
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