Log-concave Distributions Research Articles

Fast generation of order statistics

Generating a single order statistic without generating the full sample can be an important task for simulations. If the density and the CDF of the distribution are given, then it is no problem to compute the density of the order statistic. In the main theorem it is shown that the concavity properties of that density depend directly on the distribution itself. Especially for log-concave distributions, all order statistics have log-concave distributions themselves. So recently suggested automatic transformed density rejection algorithms can be used to generate single order statistics. This idea leads to very fast generators. For example for the normal and gamma distributions, the suggested new algorithms are between 10 and 60 times faster than the algorithms suggested in the literature.

Asymptotics of optimal quantizers for some scalar distributions

We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non- log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew & Wise Theorem and finally the asymptotic size of the Voronoi tessels.

Posterior Analysis of Stochastic Frontier Models with Truncated Normal Errors

Previous work in stochastic frontier models with exponentially distributed one-sided errors using both Gibbs sampling and Monte Carlo integration with importance sampling reveals the enormous computational gains that can be achieved using the former. This paper takes up inference in another interesting class of stochastic frontier models, those with truncated normal one-sided error terms, and shows that posterior simulation involves drawing from standard or log-concave distributions, implying that Gibbs sampling is an efficient solution to the Bayesian integration problem. The sampling behavior of the Bayesian procedure is investigated using a Monte Carlo experiment. The method is illustrated using US airline data.

A simple universal generator for continuous and discrete univariate T-concave distributions

We use inequalities to design short universal algorithms that can be used to generate random variates from large classes of univariate continuous or discrete distributions (including all log-concave distributions). The expected time is uniformly bounded over all these distributions. The algorithms can be implemented in a few lines of high-level language code. In opposition to other black-box algorithms hardly any setup step is required, and thus it is superior in the changing-parameter case.

Algorithm 802

Different automatic (also called universal or black-box) methods have been suggested to sample from univariate log-concave distributions. Our new automatic algorithm for bivariate log-concave distributions is based on the method of transformed density rejection. In order to construct a hat function for a rejection algorithm the bivariate density is transformed by the logarithm into a concave function. Then it is possible to construct a dominating function by taking the minimum of several tangent planes, which are by exponentiation transformed back into the original scale. The choice of the points of contact is automated using adaptive rejection sampling. This means that points that are rejected by the rejection algorithm can be used as additional points of contact. The article describes the details how this main idea can be used to construct Algorithm ALC2D that can generate random pairs from all bivariate log-concave distributions with known domain, computable density, and computable partial derivatives.

Log-concave and concave distributions in reliability

Nonparametric classes of life distributions are usually based on the pattern of aging in some sense. The common parametric families of life distributions also feature monotone aging. In this paper we consider the class of log-concave distributions and the subclass of concave distributions. The work is motivated by the fact that most of the common parametric models of life distributions (including Weibull, Gamma, log-normal, Pareto, and Gompertz distributions) are log-concave, while the remaining life of maintained and old units tend to have a concave distribution. The classes of concave and log-concave distributions do not feature monotone aging. Nevertheless, these two classes are shown to have several interesting and useful properties. We examine the closure of these classes under a number of reliability operations, and provide sharp reliability bounds for nonmaintained and maintained units having life distribution belonging to these classes. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 419–433, 1999

Log-Sobolev inequalities and sampling from log-concave distributions

We consider the problem of sampling according to a distribution with log-concave density F over a convex body $K \subseteq \mathbf{R}^n$. The sampling is done using a biased random walk and we give improved polynomial upper bounds on the time to get a sample point with distribution close to F.

A rejection technique for sampling from log-concave multivariate distributions

Different universal methods (also called automatic or black-box methods) have been suggested for sampling form univariate log-concave distributions. The descriptioon of a suitable universal generator for multivariate distributions in arbitrary dimensions has not been published up to now. The new algorithm is based on the method of transformed density rejection. To construct a hat function for the rejection algorithm the multivariate density is transformed by a proper transformation T into a concave function (in the case of log-concave density T(x) = log( x ).) Then it is possible to construct a dominating function by taking the minimum of serveral tangent hyperplanes that are transformed back by T -1 into the original scale. The domains of different pieces of the hat function are polyhedra in the multivariate case. Although this method can be shown to work, it is too slow and complicated in higher dimensions. In this article we split the ℝ n into simple cones. The hat function is constructed piecewise on each of the cones by tangent hyperplanes. The resulting function is no longer continuous and the rejection constant is bounded from below but the setup and the generation remains quite fast in higher dimensions; for example, n = 8. The article describes the details of how this main idea can be used to construct algorithm TDRMV that generates random tuples from a multivariate log-concave distribution with a computable density. Although the developed algorithm is not a real black box method it is adjustable for a large class of log-concave densities.

When Is a Renewal Process Convexly Parameterized in Its Mean Parameterization?

The convex (concave) parameterization of a generalized renewal process is considered in this paper. It is shown that if the interrenewal times have log concave distributions or have log concave survival functions (i.e., an increasing failure rate distribution), then the renewal process is convexly (concavely) parameterized in its mean parameterization.

Extremal properties of half-spaces for log-concave distributions

The isoperimetric problem for log-concave product measures in $\mathbb{R}^n$ equipped with the uniform distance is considered. Necessary and sufficient conditions under which standard half-spaces are extremal are presented.

Correction: Sampling from Log-Concave Distributions

Correction: Sampling from Log-Concave Distributions

Sampling from Log-Concave Distributions

We consider the problem of sampling according to a distribution with log-concave density $F$ over a convex body $K \subseteq \mathbb{R}^n$. The sampling is done using a biased random walk, and we prove polynomial upper bounds on the time to get a sample point with distribution close to $F$.

A universal generator for discrete log-concave distributions

We give an algorithm that can be used to sample from any discrete log-concave distribution (e.g. the binomial and hypergeometric distributions). It is based on rejection from a discrete dominating distribution that consists of parts of the geometric distribution. The algorithm is uniformly fast for all discrete log-concave distributions and not much slower than algorithms designed for a single distribution.

Near ignorance classes of log-concave priors for the location model

Near ignorance classes (NIC) for the normal location model are defined in Pericchi and Walley (1991) as a proper alternative to the improper uniform prior. In this paper we study the feasibility of such classes using log-concave distributions such as the logistic and the Box and Tiao exponential power family. We use Meeden and Isaacson’s theory of posterior moments for exponential family likelihoods, as well as results for posterior distributions under log-concave priors. The main conclusion that emerges is that exponential tails are the lightest permitted in order to produce NICs with non-vacuous posterior inference, for the normal location model.