Logarithmically concave function: Difference between revisions

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*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 >= 1) will be log-concave. This property is heavily used in general-purpose [[Gibbs sampling]] programs such as [[Bayesian inference using Gibbs sampling|BUGS]] and [[Just another Gibbs sampler|JAGS]], which are thereby able to use [[adaptive rejection sampling]] over a wide variety of [[conditional distribution]]s derived from the product of other distributions.

* If a density is log-concave, so is its [[Survival function]]<ref>See {{cite journal |first=Mark |last=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 |url=http://www.econ.ucsb.edu/~tedb/Theory/delta.pdf }}</ref>.

* If a density is log-concave, so is its [[Survival function]].

* If a density is log-concave, it has a monotone [[Hazard ratio|hazard rate]] (MHR)]], and thus 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.

::<math>\frac{d}{dx}\log\left(1-F(x)\right) = -\frac{f(x)}{1-F(x)}</math> which is increasingdecreasing as it is the derivative of a concave function.

==See also==