Peaks-over-threshold stability of multivariate generalized Pareto distributions

It is well-known that the univariate generalized Pareto distributions (GPD) are characterized by their peaks-over-threshold (POT) stability. We extend this result to multivariate GPDs.It is also shown that this POT stability is asymptotically shared by distributions which are in a certain neighborhood of a multivariate GPD. A multivariate extreme value distribution is a typical example.The usefulness of the results is demonstrated by various applications. We immediately obtain, for example, that the excess distribution of a linear portfolio with positive weights a_i, i≤d, is independent of the weights, if (U₁,…,U_d) follows a multivariate GPD with identical univariate polynomial or Pareto margins, which was established by Macke [On the distribution of linear combinations of multivariate EVD and GPD distributed random vectors with an application to the expected shortfall of portfolios, Diploma Thesis, University of Würzburg, 2004, (in German)] and Falk and Michel [Testing for tail independence in extreme value models. Ann. Inst. Statist. Math. 58 (2006) 261-290]. This implies, for instance, that the expected shortfall as a measure of risk fails in this case.     

Mixed effects regression trees for clustered data

This paper presents an extension of the standard regression tree method to clustered data. Previous works extending tree methods to accommodate correlated data are mainly based on the multivariate repeated-measures approach. We propose a “mixed effects regression tree” method where the correlated observations are viewed as nested within clusters rather than as vectors of multivariate repeated responses. The proposed method can handle unbalanced clusters, allows observations within clusters to be split, and can incorporate random effects and observation-level covariates. We implemented the proposed method using a standard tree algorithm within the framework of the expectation-maximization (EM) algorithm. The simulation results show that the proposed regression tree method provides substantial improvements over standard trees when the random effects are non negligible. A real data example is used to illustrate the method.     

Simulation of random vectors from three-dimensional spherically symmetric stable distributions

A simple approximate algorithm of sampling random vectors from three-dimensional symmetric stable distributions based on analytical approximation of inverse distribution functions is proposed. Explicit expressions for simulating the stable distributions with indices α=0.2(0.1) 1.8 are presented. The errors of the algorithm do not exceed 1–2% in the probability density function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

High-dimensional generation of Bernoulli random vectors

The objective of this paper is to explore different modeling strategies to generate high-dimensional Bernoulli vectors. We discuss the multivariate Bernoulli (MB) distribution, probe its properties and examine three models for generating random vectors. A latent multivariate normal model whose bivariate distributions are approximated with Plackett distributions with univariate normal distributions is presented. A conditional mean model is examined where the conditional probability of success depends on previous history of successes. A mixture of beta distributions is also presented that expresses the probability of the MB vector as a product of correlated binary random variables. Each method has a domain of effectiveness. The latent model offers unpatterned correlation structures while the conditional mean and the mixture model provide computational feasibility for high-dimensional generation of MB vectors.     

Simulating Multivariate Extreme Value Distributions of Logistic Type

Methods are given for simulating from symmetric and asymmetric versions of the multivariate logistic distribution, and from other multivariate extreme value distributions based on the well known logistic model. We consider two general approaches. The first approach uses transformations to derive random variables with a joint distribution function from which it is easy to simulate. The second approach derives from a specification of conditionally independent marginal components, conditioning on positive stable random variables. This specification extends to models of nested or hierarchical type and leads to an efficient way of incorporating marginal censoring. The algorithms presented in Sections 2 and 3 are available on request from the author. They are also included in the R (Ihaka and Gentleman, 1996) package evd (Stephenson, 2002), which is available from http://www.maths.lancs.ac.uk/~stephena/.     

On a new multivariate IFR ageing notion based on the standard construction

Many criteria of ageing for random variables or vectors have been proposed in the literature over many years. For instance, a random variable is increasing in failure rate (IFR) if, and only if, it can be ordered with an exponentially distributed random variable in the classical univariate convex transform order. A new multivariate generalization of the convex transform order has recently been proposed in the literature. In this work, we propose a new multivariate IFR notion for multivariate distributions based on comparisons in this new order with a properly defined exponentially distributed random vector. Properties, applications, and illustrations of this new notion are given as well. Copyright © 2016 John Wiley & Sons, Ltd.     

Some notes on multivariate generalized Pareto distributions

The investigation of multivariate generalized Pareto distributions (GPDs) has begun only recently and there are slightly varying definitions of GPDs available. In this article we investigate the one from Section 5.1 of Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004], which does not differ in the area of interest from those of other authors. We first give an interpretation of the case of independence in terms of the peaks-over-threshold approach. This case is also used in dimension d=3 by Falk et al. [Laws of Small Numbers: Extremes and Rare Events, second ed., Birkhäuser, Basel, 2004] as a counterexample to show that GP functions are not necessarily distribution functions on their entire support. We generalize this counterexample to an arbitrary dimension d≥3 and demonstrate also that other GP functions show this behavior. Finally we show that different GPDs can lead to the same conditional probability measure in the area of interest.     

Simulation of geometric stable and other limiting multivariate distributions arising in random summation scheme

A method for simulation of ν_α-stable random vectors based on their representation as location and scale mixtures of α-stable random vectors is discussed. The method has applications in modeling multivariate financial portfolios and Monte Carlo estimation procedures.     

Strong convergence of multivariate point processes of exceedances

We study the asymptotic behavior of vectors of point processes of exceedances of random thresholds based on a triangular scheme of random vectors. Multivariate maxima w.r.t. marginal ordering may be regarded as a special case. It is proven that strong convergence—that is convergence of distributions w.r.t. the variational distance—of such multivariate point processes holds if, and only if, strong convergence of multivariate maxima is valid. The limiting process of multivariate point processes of exceedances is built by a certain Poisson process. Auxiliary results concerning upper bounds on the variational distance between vectors of point processes are of interest in its own right.The author was supported by the Deutsche Forschungsgemeinschaft.     

Estimation of Parameters of the Generalized Pareto\Distribution by the Least Squares

Traditional estimations of parameters of the generalized Pareto distribution (GPD) are generally constrained by the shape parameter of GPD. Such as: the method-of-moments (MOM), the probability-weighted moments (PWM), L-moments (LM), the maximum likelihood estimation (MLE) and so on. In this paper we use the fact that GPD can be transformed into the exponential distribution and use the results of parameters estimation for the exponential distribution, than we propose parameters estimators of the two-parameter or three-parameter GPD by the least squares method. Some asymptotic results are provided and the proposed method not constrained by the shape parameter of GPD. A simulation study is carried out to evaluate the performance of the proposed method and to compare them with other methods suggested in this paper. The simulation results indicate that the proposed method performs better than others in some common situation.     

Generalized skew-elliptical distributions and their quadratic forms

This paper introduces generalized skew-elliptical distributions (GSE), which include the multivariate skew-normal, skew-t, skew-Cauchy, and skew-elliptical distributions as special cases. GSE are weighted elliptical distributions but the distribution of any even function in GSE random vectors does not depend on the weight function. In particular, this holds for quadratic forms in GSE random vectors. This property is beneficial for inference from non-random samples. We illustrate the latter point on a data set of Australian athletes.     

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.     

A new data envelopment analysis model with fuzzy random inputs and outputs

This paper first presents several formulas for mean chance distributions of triangular fuzzy random variables and their functions, then develops a new class of fuzzy random data envelopment analysis (FRDEA) models with mean chance constraints, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. According to the established formulas for the mean chance distributions, we can turn the mean chance constraints into their equivalent stochastic ones. On the other hand, since the objective in the FRDEA model is the expectation about the ratio of the weighted sum of outputs and the weighted sum of inputs for a target decision-making unite (DMU), for general fuzzy random inputs and outputs, we suggest an approximation method to evaluate the objective; and for triangular fuzzy random inputs and outputs, we propose a method to reduce the objective to its equivalent stochastic one. As a consequence, under the assumption that the inputs and the outputs are triangular fuzzy random vectors, the proposed FRDEA model can be reduced to its equivalent stochastic programming one, in which the constraints contain the standard normal distribution function, and the objective is the expectation for a function of the normal random variable. To solve the equivalent stochastic programming model, we design a hybrid algorithm by integrating stochastic simulation and genetic algorithm (GA). Finally, one numerical example is presented to demonstrate the proposed FRDEA modeling idea and the effectiveness of the designed hybrid algorithm.     

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.     

Generating multivariate correlated samples

Summary  Generating random samples from multivariate distributions is a common, requirement in many fields of study. Often the complete joint distribution is not specified to the scientist. This paper addresses the situation where only the marginals and the correlation matrix are specified. We suggest a deterministic algorithm, PERMCORR, to approximately achieve the required correlation structure that can be used to get good initial values to standard stochastic algorithms. In many situations the output of PERMCORR is already accurate enough to preempt any need for running an expensive stochastic algorithm. We provide some theoretical justification for our method as well as simulation studies. We also provide a bootstrap technique based on PERMCORR.     

Testing for a multivariate generalized Pareto distribution

It has recently been shown by Rootzén and Tajvidi (Bernoulli, 12:917–930, 2006) that modelling exceedances of a random variable over a high threshold (peaks-over-threshold approach [POT]) can also in the multivariate setup be done rationally only by a multivariate generalized Pareto distribution (GPD). The selection of a proper threshold is, however, a crucial problem. The contribution of this paper is twofold: We develop first a non asymptotic and exact level-α test based on the single-sample t-test, which checks whether multivariate data are actually generated by a multivariate GPD. Secondly, this procedure is utilized for the derivation of a t-test based threshold selection rule in multivariate peaks-over-threshold models. The application to a hydrological data set illustrates this approach.      

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 generating multivariate Poisson data in management science applications

Generating multivariate Poisson random variables is essential in many applications, such as multi echelon supply chain systems, multi‐item/multi‐period pricing models, accident monitoring systems, etc. Current simulation methods suffer from limitations ranging from computational complexity to restrictions on the structure of the correlation matrix, and therefore are rarely used in management science. Instead, multivariate Poisson data are commonly approximated by either univariate Poisson or multivariate Normal data. However, these approximations are often not adequate in practice. In this paper, we propose a conceptually appealing correction for NORTA (NORmal To Anything) for generating multivariate Poisson data with a flexible correlation structure and rates. NORTA is based on simulating data from a multivariate Normal distribution and converting it into an arbitrary continuous distribution with a specific correlation matrix. We show that our method is both highly accurate and computationally efficient. We also show the managerial advantages of generating multivariate Poisson data over univariate Poisson or multivariate Normal data. Copyright © 2011 John Wiley & Sons, Ltd.     

A randomized maximum cross-section method to simulate random structure systems with distributed transitions

In this paper, random structure systems with distributed transitions are considered. A theorem on the form of conditional structure distributions is proved. To simulate these distributions, a statistical algorithm using a randomized method of maximum cross-section is constructed. Also, a modified version of this algorithm using simulation with a single random number is constructed. The algorithms are used to simulate the numerical solution of random structure systems with distributed transitions. A theorem on weak convergence of a numerical solution obtained by the algorithms is proved.     

Multivariate analysis of variance with fewer observations than the dimension

In this article, we consider the problem of testing a linear hypothesis in a multivariate linear regression model which includes the case of testing the equality of mean vectors of several multivariate normal populations with common covariance matrix Σ, the so-called multivariate analysis of variance or MANOVA problem. However, we have fewer observations than the dimension of the random vectors. Two tests are proposed and their asymptotic distributions under the hypothesis as well as under the alternatives are given under some mild conditions. A theoretical comparison of these powers is made.