Construction of asymmetric multivariate copulas

In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.     

Perfect simulation with exponential tails on the running time

Monte Carlo algorithms typically need to generate random variates from a probability distribution described by an unnormalized density or probability mass function. Perfect simulation algorithms generate random variates exactly from these distributions, but have a running time T that is itself an unbounded random variable. This article shows that commonly used protocols for creating perfect simulation algorithms, such as Coupling From the Past can be used in such a fashion that the running time is unlikely to be very much larger than the expected running time. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Generating random correlation matrices based on vines and extended onion method

We extend and improve two existing methods of generating random correlation matrices, the onion method of Ghosh and Henderson [S. Ghosh, S.G. Henderson, Behavior of the norta method for correlated random vector generation as the dimension increases, ACM Transactions on Modeling and Computer Simulation (TOMACS) 13 (3) (2003) 276–294] and the recently proposed method of Joe [H. Joe, Generating random correlation matrices based on partial correlations, Journal of Multivariate Analysis 97 (2006) 2177–2189] based on partial correlations. The latter is based on the so-called D-vine. We extend the methodology to any regular vine and study the relationship between the multiple correlation and partial correlations on a regular vine. We explain the onion method in terms of elliptical distributions and extend it to allow generating random correlation matrices from the same joint distribution as the vine method. The methods are compared in terms of time necessary to generate 5000 random correlation matrices of given dimensions.     

Interpretations of some parameter dependent generalizations of classical matrix ensembles

Two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions (PDFs). As the parameter is varied, one interpolates between the eigenvalue PDF for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue PDF for a single classical ensemble with unitary symmetry, while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry. We give interpretations of these PDFs in terms of probabilities associated to the continuous Robinson-Schensted-Knuth correspondence between matrices, with entries chosen from certain exponential distributions, and non-intersecting lattice paths, and in the course of this probability measures on partitions and pairs of partitions are identified. The latter are generalized by using Macdonald polynomial theory, and a particular continuum limit – the Jacobi limit – of the resulting measures is shown to give PDFs related to those appearing in the work of Anderson on the Selberg integral, and also in some classical work of Dixon. By interpreting Andersons and Dixons work as giving the PDF for the zeros of a certain rational function, it is then possible to identify random matrices whose eigenvalue PDFs realize the original parameter dependent PDFs. This line of theory allows sampling of the original parameter dependent PDFs, their Dixon-Anderson-type generalizations and associated marginal distributions, from the zeros of certain polynomials defined in terms of random three term recurrences.Supported by the Australian Research Council     

Optimizing random scan Gibbs samplers

The Gibbs sampler is a popular Markov chain Monte Carlo routine for generating random variates from distributions otherwise difficult to sample. A number of implementations are available for running a Gibbs sampler varying in the order through which the full conditional distributions used by the Gibbs sampler are cycled or visited. A common, and in fact the original, implementation is the random scan strategy, whereby the full conditional distributions are updated in a randomly selected order each iteration. In this paper, we introduce a random scan Gibbs sampler which adaptively updates the selection probabilities or “learns” from all previous random variates generated during the Gibbs sampling. In the process, we outline a number of variations on the random scan Gibbs sampler which allows the practitioner many choices for setting the selection probabilities and prove convergence of the induced (Markov) chain to the stationary distribution of interest. Though we emphasize flexibility in user choice and specification of these random scan algorithms, we present a minimax random scan which determines the selection probabilities through decision theoretic considerations on the precision of estimators of interest. We illustrate and apply the results presented by using the adaptive random scan Gibbs sampler developed to sample from multivariate Gaussian target distributions, to automate samplers for posterior simulation under Dirichlet process mixture models, and to fit mixtures of distributions.     

Multiway Dependence in Exponential Family Conditional Distributions

Conditionally specified statistical models are frequently constructed from one-parameter exponential family conditional distributions. One way to formulate such a model is to specify the dependence structure among random variables through the use of a Markov random field (MRF). A common assumption on the Gibbsian form of the MRF model is that dependence is expressed only through pairs of random variables, which we refer to as the “pairwise-only dependence” assumption. Based on this assumption, J. Besag (1974, J. Roy. Statist. Soc. Ser. B36, 192–225) formulated exponential family “auto-models” and showed the form that one-parameter exponential family conditional densities must take in such models. We extend these results by relaxing the pairwise-only dependence assumption, and we give a necessary form that one-parameter exponential family conditional densities must take under more general conditions of multiway dependence. Data on the spatial distribution of the European corn borer larvae are fitted using a model with Bernoulli conditional distributions and several dependence structures, including pairwise-only, three-way, and four-way dependencies.     

A Heuristic for Moment-Matching Scenario Generation

In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discrete joint distribution consistent with specified values of the first four marginal moments and correlations. The joint distribution is constructed by decomposing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations to achieve the correct correlations without changing the marginal moments.With the algorithm, we can generate 1000 one-period scenarios for 12 random variables in 16 seconds, and for 20 random variables in 48 seconds, on a Pentium III machine.     

Reduced-rank regression for the multivariate linear model

The problem of estimating the regression coefficient matrix having known (reduced) rank for the multivariate linear model when both sets of variates are jointly stochastic is discussed. We show that this problem is related to the problem of deciding how many principal components or pairs of canonical variates to use in any practical situation. Under the assumption of joint normality of the two sets of variates, we give the asymptotic (large-sample) distributions of the various estimated reduced-rank regression coefficient matrices that are of interest. Approximate confidence bounds on the elements of these matrices are then suggested using either the appropriate asymptotic expressions or the jackknife technique.     

A heuristic approach for the generation of multivariate random samples with specified marginal distributions and correlation matrix

Summary  This paper presents a heuristic approach for multivariate random number generation. Our aim is to generate multivariate samples with specified marginal distributions and correlation matrix, which can be incorporated into risk analysis models to conduct simulation studies. The proposed sampling approach involves two distinct steps: first a univariate random sample from each specified probability distribution is generated; then a heuristic combinatorial optimization procedure is used to rearrange the generated univariate samples, in order to obtain the desired correlations between them. The combinatorial optimization step is performed with a simulated annealing algorithm, which changes only the positions and not the values of the numbers generated in the first step. The proposed multivariate sampling approach can be used with any type of marginal distributions: continuous or discrete, parametric or non-parametric, etc.     

Short universal generators via generalized ratio-of-uniforms method

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 for a particular generator. The algorithms can be implemented in a few lines of high level language code.

     

Uniformly distributed sums of bernoulli variables

Sums of independent, identically distributed (iid) binomial variates have binomial distributions; yet it is possible to construct a sequence of binomial distributions over {0, 1} for variatesX ₁,X ₂, ... such that all partial sumsY _i =X ₁ + ... +X _i have uniform distributions. The price to pay is to give up the iid condition. Requiring the property of only one sum does not alleviate the situation much.It is also possible to generate on a computerm × n-matrices, of 0–1 binomial variates with uniformly distributed row and column sums of all major submatrices, but only for smallm andn. Even a three-dimensional 2 × 2 × 2 array can have a similar property.Other target distributions than the rectangular are possible, but cumbersome. An example with smaller variance is given.The results were needed for simulating the performance of some Operations Research algorithms.Dedicated to Peter Naur on the occasion of his 60th birthday     

Extensions of Wilks' integral equations and distributions of test statistics

Summary Wilks [26] introduced two integral equations in connection with distribution problems in statistics. He called them Type A and Type B equations. Tretter and Walster ([22], [24]) solved the Type B equation and obtained the null and non-null distributions of the likelihood ratio criterion for testing linear hypotheses in the multinormal case. In this article we present several types of solutions of these equations along with new equations called Types C, D, E and F with their solutions. These include the integral equations satisfied by the density of a random variable which is (a) product of independent real gamma variates; (b) products of independent real beta variates; (c) ratio of products of independent beta and gamma variates; (d) arbitrary powers of products of gamma and beta variates; (e) arbitrary powers of products and ratios of beta and gamma variates, and more general cases.     

Random sets. A survey of results and applications

A survey of current directions in the theory of random closed sets is presented; these include: the central limit theorem, the law of large numbers for Minkowski sums and unions of random sets, semi-Markov random closed sets, Boolean models and statistical estimation of their parameters, specification of distributions and associated problems of capacity theory. Weak convergence of random closed sets is defined and its application to limit theorems for graphs and epi-graphs of random processes and problems of stochastic optimization is described. Other connections with the theory of random processes (level sets, multivalued and controllable random processes) are also discussed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 12, pp. 1587–1599, December, 1991.     

On the Convergence of Successive Substitution Sampling

Abstract

The problem of finding marginal distributions of multidimensional random quantities has many applications in probability and statistics. Many of the solutions currently in use are very computationally intensive. For example, in a Bayesian inference problem with a hierarchical prior distribution, one is often driven to multidimensional numerical integration to obtain marginal posterior distributions of the model parameters of interest. Recently, however, a group of Monte Carlo integration techniques that fall under the general banner of successive substitution sampling (SSS) have proven to be powerful tools for obtaining approximate answers in a very wide variety of Bayesian modeling situations. Answers may also be obtained at low cost, both in terms of computer power and user sophistication. Important special cases of SSS include the “Gibbs sampler” described by Gelfand and Smith and the “IP algorithm” described by Tanner and Wong. The major problem plaguing users of SSS is the difficulty in ascertaining when “convergence” of the algorithm has been obtained. This problem is compounded by the fact that what is produced by the sampler is not the functional form of the desired marginal posterior distribution, but a random sample from this distribution. This article gives a general proof of the convergence of SSS and the sufficient conditions for both strong and weak convergence, as well as a convergence rate. We explore the connection between higher-order eigenfunctions of the transition operator and accelerated convergence via good initial distributions. We also provide asymptotic results for the sampling component of the error in estimating the distributions of interest. Finally, we give two detailed examples from familiar exponential family settings to illustrate the theory.     

The problem of identification of parameters by the distribution of the maximum random variable

Suppose that X₁, X₂,…, X_n are independently distributed according to certain distributions. Does the distribution of the maximum of {X₁, X₂,…, X_n} uniquely determine their distributions? In the univariate case, a general theorem covering the case of Cauchy random variables is given here. Also given is an affirmative answer to the above question for general bivariate normal random variables with non-zero correlations. Bivariate normal random variables with nonnegative correlations were considered earlier in this context by T. W. Anderson and S. G. Ghurye.     

General canonical correlations with applications to group symmetry models

In this paper, we define general canonical correlations, which generalize the canonical correlations developed by Hotelling, and general canonical covariate pairs, the corresponding linear statistic. We also define canonical variance distances with corresponding canonical distance variates. In a rather broad setting, these parameters and their corresponding linear statistics are characterized in terms of certain eigenvalues and eigenvectors. For seven of the ten group symmetry testing problems discussed in Andersson, Brøns, and Jensen (1983) [4], these are the eigenvalues used to represent the maximal invariant statistic, and additional observations regarding the canonical correlations are made for these testing problems.     

Two characteristic properties of normal distributions in the class NH

In this note are deduced two characterizations of normal distributions in the class NH of probability distributions on a plane whose densities admit diagonal expansions in series of Hermite polynomials and the marginal distributions are standardized and normal.Translated from Matematicheskie Zametkl, Vol. 20, No. 1, pp. 139–146, July, 1976.     

Representing and Learning Distributions with the Aid of a Neural Network

This paper describes a method by which a neural network learns to fit a distribution to sample data. The neural network may be used to replace the input distributions required in a simulation or mathematical model and it allows random variates to be generated for subsequent use in the model. Results are given for several data sets which indicate the method is robust and can represent different families of continuous distributions. The neural network is a three-layer feed-forward network of size (1-3-3-1). This paper suggests that the method is an alternative approach to the problem of selection of suitable continuous distributions and random variate generation techniques for use in simulation and mathematical models.     

Theory of the conductivity of ordered alloys

A theory of the electrical conductivity of alloys going beyond the coherent potential approximation and taking into account statistical correlations in the scattering of the electrons by the atoms is developed. The two-particle Green's functions for the electrons is calculated with allowance for scattering by pairs of atoms (the zeroth approximation corresponds to the coherent potential approximation), the correlations being described by means of the parameters of the short-and long-range order. It is shown that the change in the electron spectrum on ordering leads to a significant change of the alloy conductivity.Institute of Metal Physics, Ukrainian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 2, pp. 279–293, May 1992.     

Limiting distributions and estimation in generalized Polya random walk scheme

Asymptotic behavior of distributions generated by Polya random walks is investigated. Unbiased estimators of these distributions are constructed for closed first-arrival plans.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 20–36, 1986.