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.     

Mixture Modeling for Longitudinal Data

In this article, we propose an unbiased estimating equation approach for a two-component mixture model with correlated response data. We adapt the mixture-of-experts model and a generalized linear model for component distribution and mixing proportion, respectively. The new approach only requires marginal distributions of both component densities and latent variables. We use serial correlations from subjects’ subgroup memberships, which improves estimation efficiency and classification accuracy, and show that estimation consistency does not depend on the choice of the working correlation matrix. The proposed estimating equation is solved by an expectation-estimating-equation (EEE) algorithm. In the E-step of the EEE algorithm, we propose a joint imputation based on the conditional linear property for the multivariate Bernoulli distribution. In addition, we establish asymptotic properties for the proposed estimators and the convergence property using the EEE algorithm. Our method is compared to an existing competitive mixture model approach in both simulation studies and an election data application. Supplementary materials for this article are available online.     

Bayesian Multivariate Distributional Regression With Skewed Responses and Skewed Random Effects

The normal and the t distribution are classical tools for building random effects regression models where both can be used for the specification of either the conditional response distribution or the random effects distribution. However, the underlying assumption of symmetry can be questionable in many applications. We, therefore, propose regression models where the skew-normal and skew-t distribution are considered for both the response and the random effects specification and embed these models in the framework of distributional regression such that regression predictors can be specified for all distributional parameters. The distributional regression framework also allows us to consider multivariate versions of the skew-normal and the skew-t distribution. For Bayesian inference, we adapt iteratively weighted least-square proposals within Markov chain Monte Carlo simulations such that they can also facilitate the inclusion of nonnormal random effects specifications. Model choice is based on the Watanabe–Akaike information criterion, in particular, to differentiate between skew and nonskew distributional specifications in a number of simulation studies. Finally, to illustrate their practical applicability, the developed models are applied to a study on cholesterol levels originating from the Framingham Heart Study and a dataset from the Demographic and Health Surveys on undernutrition among children in Nigeria. Supplementary material for this article is available online.     

A Note on Multivariate Distributions with Specified Marginals

Johnson and Tenenbein describe a procedure for generating random values from a bivariate distribution with specified marginal forms and any required correlation between the two variables. This note shows that the approach can be extended to produce values from general multivariate distributions in a straightforward manner.     

A new construction for skew multivariate distributions

This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal (type A and type B) and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions.     

On approximating max-stable processes and constructing extremal copula functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical inferences, the paper focuses on a family of sectional multivariate extreme value copula (SMEVC) functions which is derived from the joint distribution functions of M4 processes. A new non-standard parameter estimation procedure is introduced, which is based on order statistics of ratios of (transformed) marginal unit Fréchet random variables, and is shown via simulation to be more efficient than a semi-parametric estimation procedure. In real data analysis, empirical results show that SMEVCs are more flexible for modeling various dependence structures, and perform better than the widely used Gumbel-Hougaard copulas.     

Nonparametric estimation of the spectral measure,and associated dependence measures,for multivariate extreme values using a limiting conditional representation

The traditional approach to multivariate extreme values has been through the multivariate extreme value distribution G, characterised by its spectral measure H and associated Pickands’ dependence function A. More generally, for all asymptotically dependent variables, H determines the probability of all multivariate extreme events. When the variables are asymptotically dependent and under the assumption of unit Fréchet margins, several methods exist for the estimation of G, H and A which use variables with radial component exceeding some high threshold. For each of these characteristics, we propose new asymptotically consistent nonparametric estimators which arise from Heffernan and Tawn’s approach to multivariate extremes that conditions on variables with marginal values exceeding some high marginal threshold. The proposed estimators improve on existing estimators in three ways. First, under asymptotic dependence, they give self-consistent estimators of G, H and A; existing estimators are not self-consistent. Second, these existing estimators focus on the bivariate case, whereas our estimators extend easily to describe dependence in the multivariate case. Finally, for asymptotically independent cases, our estimators can model the level of asymptotic independence; whereas existing estimators for the spectral measure treat the variables as either being independent, or asymptotically dependent. For asymptotically dependent bivariate random variables, the new estimators are found to compare favourably with existing estimators, particularly for weak dependence. The method is illustrated with an application to finance data.     

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.     

Simulation of certain multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently. For further progress with these distributions simulation methods are an important part. We describe several methods of simulating GPDs, beginning with an efficient method for the logistic GPD. The algorithm is based on the Shi transformation, which was already used for the simulation of multivariate extreme value distributions (EVDs) of logistic type. In the sequel another algorithm is presented simulating a broader class of GPDs. Due to its numerical complexity it is only practicably applicable in low dimensions. A method is given to generate unconditional GPD random vectors from conditionally GPD distributed random vectors. A short application of the simulation methods in the analysis of a real hydrological data set concludes the article. The simulation algorithms are available on the author’s home page .      

An alternative method for computing mean and covariance matrix of some multivariate distributions

Computing the mean and covariance matrix of some multivariate distributions, in particular, multivariate normal distribution and Wishart distribution are considered in this article. It involves a matrix transformation of the normal random vector into a random vector whose components are independent normal random variables, and then integrating univariate integrals for computing the mean and covariance matrix of a multivariate normal distribution. Moment generating function technique is used for computing the mean and covariances between the elements of a Wishart matrix. In this article, an alternative method that uses matrix differentiation and differentiation of the determinant of a matrix is presented. This method does not involve any integration.     

Multivariate extreme value distributions for stationary Gaussian sequences

Under weak regularity conditions of the covariance sequence, it is shown that the joint limiting distribution of the maxima on each coordinate of a stationary Gaussian multivariate sequence is that of independent random variables with marginal Gumbel distributions.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

Modeling clustered binary data with nonparametric unobserved heterogeneity: An application to stock crash analysis

Various random effects models have been developed for clustered binary data; however, traditional approaches to these models generally rely heavily on the specification of a continuous random effect distribution such as Gaussian or beta distribution. In this article, we introduce a new model that incorporates nonparametric unobserved random effects on unit interval (0,1) into logistic regression multiplicatively with fixed effects. This new multiplicative model setup facilitates prediction of our nonparametric random effects and corresponding model interpretations. A distinctive feature of our approach is that a closed-form expression has been derived for the predictor of nonparametric random effects on unit interval (0,1) in terms of known covariates and responses. A quasi-likelihood approach has been developed in the estimation of our model. Our results are robust against random effects distributions from very discrete binary to continuous beta distributions. We illustrate our method by analyzing recent large stock crash data in China. The performance of our method is also evaluated through simulation studies.     

On testing equality of pairwise rank correlations in a multivariate random vector

Spearman’s rank-correlation coefficient (also called Spearman’s rho) represents one of the best-known measures to quantify the degree of dependence between two random variables. As a copula-based dependence measure, it is invariant with respect to the distribution’s univariate marginal distribution functions. In this paper, we consider statistical tests for the hypothesis that all pairwise Spearman’s rank correlation coefficients in a multivariate random vector are equal. The tests are nonparametric and their asymptotic distributions are derived based on the asymptotic behavior of the empirical copula process. Only weak assumptions on the distribution function, such as continuity of the marginal distributions and continuous partial differentiability of the copula, are required for obtaining the results. A nonparametric bootstrap method is suggested for either estimating unknown parameters of the test statistics or for determining the associated critical values. We present a simulation study in order to investigate the power of the proposed tests. The results are compared to a classical parametric test for equal pairwise Pearson’s correlation coefficients in a multivariate random vector. The general setting also allows the derivation of a test for stochastic independence based on Spearman’s rho.     

A simple construction of optimal estimation in multivariate marginal Cox regression

The multivariate extension of the Cox model proposed by Wei,Lin and Weissfeld in 1989 has been widely used for analyzing multivariate survival data.Under the model assumption,failure times from an individual are assumed to marginally follow their respective proportional hazards regression relation,leaving the joint distribution completely unspecified.This paper presents a simple approach to efficiency improvement through segmentation of stochastic integrals in the marginal estimating equations and incorporation of the limiting covariance structure.It is shown that when partition of the time interval is done at a suitable rate,the resulting estimator is consistent and asymptotically normal.Through the reproducing kernel Hilbert space arising from the covariance function of the limiting Gaussian process,it is also shown that the proposed estimator is asymptotically optimal within a reasonable class of estimators under marginal specification.Simulations are conducted to assess the finite-sample performance of the proposed method.     

封闭型保单组未来损失现值向量的联合分布及渐近分布

本文针对封闭型保单组,利用历年死亡人数随机向量D,将保单组的未来给付现值随机变量和未来损失现值随机变量表达为某个满秩矩阵和D的乘积,根据D服从多项分布的性质,得到未来损失现值随机向量渐近服从多元正态分布的结果,为分析责任准备金提供了一个新的框架.     

Multivariate Lukacs theorem

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.     

On the logistic midrange

Summary It is well-known that for a large family of distributions, the sample midrange is asymptotically logistic. In this article, the logistic midrange is closely examined. Its distribution function is derived using Dixon's formula (Bailey (1935,Generalized Hypergeometric Series, Cambridge University Press, p. 13)) for the generalized hypergeometric function with unit argument, together with appropriate techniques for the inversion of (bilateral) Laplace transforms. Several relationships in distribution are established between the midrange and sample median of the logistic and Laplace random variables. Possible applications in testing for outliers are also discussed.     

Distribution modeling for reliability analysis: Impact of multiple dependences and probability model selection

Reliability analysis requires modeling of joint probability distribution of uncertain parameters, which can be a challenge since the random variables representing the parameter uncertainties may be correlated. For convenience, a Gaussian data dependence is commonly assumed for correlated random variables. This paper first investigates the effect of multidimensional non-Gaussian data dependences underlying the multivariate probability distribution on reliability results. Using different bivariate copulas in a vine structure, various data dependences can be modeled. The associated copula parameters are identified from available statistical information by moment matching techniques. After the development of the vine copula model for representing the multivariate probability distribution, the reliability involving correlated random variables is evaluated based on the Rosenblatt transformation. The impact of data dependence is significant because a large deviation in failure probability is observed, which emphasizes the need for accurate dependence characterization. A practical method for dependence modeling based on limited data is thus provided. The result demonstrates that the non-Gaussian data dependences can be real in practice, and the reliability can be biased if the Gaussian dependence is used inappropriately. Moreover, the effect of conditioning order on reliability should not be overlooked except that the vine structure contains only one type of copula.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.