(For even sample size n this is an approximation of the real density, since then the median is the mean of two random variates.) Using the theorem that every marginal distribution of a log-concave multivariate distribution is again log-concave [Prekopa 1973, Theorem 8] we can conclude that the densities of every order statistics of a log-concave distribution is again log-concave (see also Hormann et al.
Generating the Median of a Sample of Size n = 199 of Exponential Power Distributed Random Variates ([Tau] [element of] [1,3]).
The break-even point is given by 3-4 random variates.
Random variates inside the squeeze are generated by mere inversion, and therefore in opposition to any other ratio-of-uniforms method, the expected number of uniform random numbers is less than two.
the method is close to inversion, and thus the resulting random variates can be used for variance reduction techniques.
Thus, for a ratio |[P.sup.s]|/|[P.sup.e]| close to 1 we have almost inversion for generating random variates. The inversion method has two advantages and is thus favored by the simulation community (see Bratley et al.
The mathematics necessary for a sweep-plane algorithm to generate uniform random variates over simple polytopes in high dimensions was collected in Leydold and Hormann [1998].
Given a computable, log-concave density f(x, y) and its partial derivatives and the domain of the distribution the algorithm can generate random variates (X, Y) with the desired distribution almost as fast as independent normal pairs can be generated by the Box-Muller method.
(Note that even for an unbounded polygon for any given x the domain of Y is bounded, if the volume below the hat is bounded.) Thus it is easy to generate random variates with density proportional to h(x, y) over any of the regions by using the conditional distribution method.
"Poisson Random Variate Generation," Research Memorandum 81-4, Purdue University, 1981.
There are many methods for generating random variates from the binomial probability mass function
The most basic Bernoulli method generates n independent U (0, 1) random variates, and returns the numbers that are less than or equal to p.