Multivariate Exponential Distribution Research Articles

An alternative multivariate exponential power distribution

In the existing multivariate Exponential Power Distribution (EPD), the shape parameter has the tendency to affect the variability in the data leading to distorted features of the distribution. This study resolves this problem by obtaining a generalized expression in place of the constant coefficient (0.5) of the quadratic form of the multivariate EPD as a function of the shape parameter, such that the variance-covariance matrix of the data is exactly equal to the scale parameter. The density function thus obtained is a more generalized multivariate EPD that provides flexibility in depicting the true underlying characteristics of the distribution for all sample sizes.

A novel multivariate copula of Raftery type with multiple dependence parameters and its neutrosophic application in finance

This paper introduces an innovative multivariate exponential distribution, specifically of Raftery type, characterized by heterogeneous dependence parameters. Various properties of this distribution family are thoroughly investigated, with particular emphasis placed on the copula derived from this model. Notably, this copula is non-exchangeable and demonstrates multiple dependence parameters. Different properties associated with this novel copula, including the examination of estimation parameters, have been thoroughly investigated. The efficacy of the proposed copula is demonstrated through its successful application in modeling a real neutrosophic dataset associated with the New York and American Stock Exchanges.

Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes

We establish a one-to-one correspondence between (i) exchangeable sequences of random variables whose finite-dimensional distributions are minimum (or maximum) infinitely divisible and (ii) non-negative, non-decreasing, infinitely divisible stochastic processes. The exponent measure of an exchangeable minimum infinitely divisible sequence is shown to be the sum of a very simple “drift measure” and a mixture of product probability measures, which uniquely corresponds to the Lévy measure of a non-negative and non-decreasing infinitely divisible process. The latter is shown to be supported on non-negative and non-decreasing functions. In probabilistic terms, the aforementioned infinitely divisible process is equal to the conditional cumulative hazard process associated with the exchangeable sequence of random variables with minimum (or maximum) infinitely divisible marginals. Our results provide an analytic umbrella which embeds the de Finetti subfamilies of many interesting classes of multivariate distributions, such as exogenous shock models, exponential and geometric laws with lack-of-memory property, min-stable multivariate exponential and extreme-value distributions, as well as reciprocal Archimedean copulas with completely monotone generator and Archimedean copulas with log-completely monotone generator.

Implementation of Multivariate Exponential Power Distribution in Discrimination and Classification of Psychological Data and Other applications

Recent advances have shown that some multivariate psychological data are deviating from usual normal assumption either in the tails or kurtosis. Thereby, allowing the call for modelling of such data using more robust elliptically contoured density which includes the normal distribution as a special case. This allowed more flexibility at the kurtosis and tail regions, which is better in handling non-normality in data analysis and also lower the cost of misclassification. The present study employed a robust model for such cases in the context of discrimination and classification of multivariate psychological disorder data using multivariate exponential distribution as an underlining model. Parameters were estimated using the method of maximum likelihood estimation and the discrimination and classification were based on the log likelihood ratio approach. The resulting models relied solidly on the shape parameter, which regulate the tails and the kurtosis, thereby  allowed flexibility. This method enable us to lower the cost of misclassification. Some other areas of applications were also considered in the paper.

A parallel algorithm for ridge-penalized estimation of the multivariate exponential family from data of mixed types

Computationally efficient evaluation of penalized estimators of multivariate exponential family distributions is sought. These distributions encompass among others Markov random fields with variates of mixed type (e.g., binary and continuous) as special case of interest. The model parameter is estimated by maximization of the pseudo-likelihood augmented with a convex penalty. The estimator is shown to be consistent. With a world of multi-core computers in mind, a computationally efficient parallel Newton–Raphson algorithm is presented for numerical evaluation of the estimator alongside conditions for its convergence. Parallelization comprises the division of the parameter vector into subvectors that are estimated simultaneously and subsequently aggregated to form an estimate of the original parameter. This approach may also enable efficient numerical evaluation of other high-dimensional estimators. The performance of the proposed estimator and algorithm are evaluated and compared in a simulation study. Finally, the presented methodology is applied to data of an integrative omics study.

Discrimination and Classification model from Multivariate Exponential Power Distribution

It is common to assume a normal distribution when discriminating and classifying a multivariate data based on some attributes. But when such data is lighter or heavier in both tails than the normal distribution, then the  probability of misclassification becomes higher giving unreliable result. This study proposed multivariate exponential power distribution a family of elliptically contoured model as underlining model for discrimination and classification. The distribution has a shape parameter which regulate the tail of the symmetric distribution to mitigate the problem of both lighter and heavier tails data, this generalizes the normal distribution and thus will definitely gives a lower misclassification error in discrimination and classification. The resulting discriminant model was compared with fisher linear discriminant function when applying to real data.

Hierarchical copulas with Archimedean blocks and asymmetric between-block pairs

A new class of hierarchical copulas is introduced based on joint survival functions of multivariate exponential mixture distributions. The key element of this construction is the mixing random vector defined by convolutions associated to a Lévy subordinator, and leading to hierarchical copulas with Archimedean within-block copulas and asymmetric between-block pair-copulas. For the specific case of two-level trees, dependence properties of pairs are investigated, and a full estimation procedure is proposed for the tree structure and parameters of the hierarchical copulas. The efficiency of the procedure is illustrated through three simulation examples and a study with two real datasets.

A multivariate CUSUM control chart for monitoring Gumbel's bivariate exponential data

AbstractExponentially distributed data are commonly encountered in high‐quality processes. Control charts dedicated to the univariate exponential distribution have been extensively studied by many researchers. In this paper, we investigate a multivariate cumulative sum (MCUSUM) control chart for monitoring Gumbel's bivariate exponential (GBE) data. Some tables are provided to determine the optimal design parameters of the proposed MCUSUM GBE chart. Furthermore, both zero‐state and steady‐state properties of the proposed MCUSUM GBE chart for the raw and the transformed GBE data are compared with the multivariate exponentially weighted moving average (MEWMA) chart and the paired individual cumulative sum (CUSUM) chart. The results show that the proposed MCUSUM GBE chart outperforms the other two types of control charts for most shift domains. In addition, an extension to Gumbel's multivariate exponential (GME) distribution is also investigated. Finally, an illustrative example is provided in order to explain how the proposed MCUSUM GBE chart can be implemented in practice.

Inferences from Asymmetric Multivariate Exponential Power Distribution

In this paper, we develop the asymptotic distribution of the covariance structure of an asymmetric multivariate exponential power distribution (AMEPD) by extending the existing Hotelling t2 distribution to its generalized form. The obtained results ‘generalised Hotelling t2 test statistic’, accommodates for the existing Mahalanobis distance between two multivariate data sets. The hazard rate and the characteristics function of the desired test statistics are established. The study further establishes the test of hypothesis and simultaneous confidence interval for the population mean vectors of p th attributes from AMEPD by using the Neymar-Pearson lemma and Pivotal quantity approaches. The obtained results, which are dependent on shape parameter flexibility and skewness parameter, generalize the test of hypothesis and simultaneous confidence intervals of various distributions which are members of the exponential power family. Simulated and real life data were used to demonstrate the usefulness of the obtained results.

How well do SEM algorithms imitate EM algorithms? A non-asymptotic analysis for mixture models

In this paper, we present a theoretical and an experimental comparison of EM and SEM algorithms for different mixture models. The SEM algorithm is a stochastic variant of the EM algorithm. The qualitative intuition behind the SEM algorithm is simple: If the number of observations is large enough, then we expect that an update step of the stochastic SEM algorithm is similar to the corresponding update step of the deterministic EM algorithm. In this paper, we quantify this intuition. We show that with high probability the update equations of any EM-like algorithm and its stochastic variant are similar, given that the input set satisfies certain properties. For instance, this result applies to the well-known EM and SEM algorithm for Gaussian mixture models and EM-like and SEM-like heuristics for multivariate power exponential distributions. Our experiments confirm that our theoretical results also hold for a large number of successive update steps. Thereby we complement the known asymptotic results for the SEM algorithm. We also show that, for multivariate Gaussian and multivariate Laplacian mixture models, an update step of SEM runs nearly twice as fast as an EM update set.

Composite likelihood estimation method for hierarchical Archimedean copulas defined with multivariate compound distributions

We consider the family of hierarchical Archimedean copulas obtained from multivariate exponential mixture distribution through compounding, as introduced by Cossette et al. (2017). We investigate ways of determining the structure of these copulas and estimating their parameters. An agglomerative clustering technique based on the matrix of Spearman’s rhos, combined with a bootstrap procedure, is used to identify the tree structure. Parameters are estimated through a top-down composite likelihood. The validity of the approach is illustrated through two simulation studies in which the procedure is explained step by step. The composite likelihood method is also compared to the full likelihood method in a simple case where the latter is computable.

On total positivity of exchangeable random variables obtained by symmetrization, with applications to failure-dependent lifetimes

Necessary and sufficient conditions for multivariate total positivity of order 2 (MTP2) for density functions of some class of exchangeable random variables are obtained. The considered densities occur via symmetrization of particular ordered random variables. As an example, a characterization of the MTP2 property for the Freund–Weinman multivariate exponential distribution is given. Furthermore, the results are applied to general failure-dependent component lifetimes in systems based on sequential order statistics. In the latter setting, the hazard rate increasing upon failure (HIF) property is also characterized. In particular, the case of underlying distributions satisfying the proportional hazards assumption is considered. The results are supplemented by an analysis of the covariances of the above multivariate exponential distribution.

A form of multivariate generalised exponential family of distribution

David Sam et.al(2012) proposed a Generalised Multivariate additive exponential distribution and further David Sam et.al(2014) extend the work by proposing Multivariate Generalized double exponential family of distribution of kind-1. In this paper, the authors bring out a family of exponential distributions in a generalized multivariate form which leads to explore more than 25 special cases of generalized multivariate exponential distribution. In subsequent sections, authors proposed distribution and its special cases are discussed.

Multivariate reliability modelling based on dependent dynamic shock models

• This paper develops new classes of multivariate reliability models with positive dependence. • The developed novel methodology is also used to generate multivariate exponential distributions. • Multivariate dependence and ageing properties of the developed models are discussed. • This paper uses advanced shock models to generate multivariate reliability models. In this paper, we stochastically model positively dependent multivariate reliability distributions based on stochastically dependent dynamic shock models. In the first part, we consider a shock model with delayed failures. This shock model will be used to construct a class of absolutely continuous multivariate reliability distributions. Explicit parametric forms for the multivariate reliability functions are suggested. Multivariate ageing properties and dependence structures of the class are discussed as well. In the second part, we obtain two types of absolutely continuous multivariate exponential distributions based on further generalized shock models.

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.

Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk

Abstract In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

A natural parametrization of multivariate distributions with limited memory

A unified formulation of the theory of d -variate wide-sense geometric ( G d W ) and Marshall–Olkin exponential ( MO d ) distributions is presented in which d -monotone set functions occupy a central role. A semi-analytical derivation of G d W and MO d distributions is deduced directly from the lack-of-memory property. In this context, the distributions are parametrized with d -monotone and d -log-monotone set functions arising from the univariate marginal distributions of minima and the d -decreasingness of the survival functions. In addition, a one-to-one correspondence is established between d -monotone (resp. d -log-monotone) set functions and d -variate (resp. d -variate min-infinitely divisible) Bernoulli distributions. The advantage of such a parametrization is that it makes the distributions highly tractable. As a showcase, we derive new results on the minimum stability and divisibility of the G d W family, and on the marginal equivalence in minima of G d W and distributions with geometric minima. Similarly, a surprisingly simple proof is given of the prominent result of Esary and Marshall (1974) on the marginal equivalence in minima of multivariate exponential distributions.

Sufficient Reductions in Regressions With Exponential Family Inverse Predictors

ABSTRACTWe develop methodology for identifying and estimating sufficient reductions in regressions with predictors that, given the response, follow a multivariate exponential family distribution. This setup includes regressions where predictors are all continuous, all categorical, or mixtures of categorical and continuous. We derive the minimal sufficient reduction of the predictors and its maximum likelihood estimator by modeling the conditional distribution of the predictors given the response. Whereas nearly all extant estimators of sufficient reductions are linear and only partly capture the sufficient reduction, our method is not limited to linear reductions. It also provides the exact form of the sufficient reduction, which is exhaustive, its maximum likelihood (ML) estimates via an iterated reweighted least-square (IRLS) estimation algorithm, and asymptotic tests for the dimension of the regression. Supplementary materials for this article are available online.

The Monge–Ampère equation in transformation theory and an application to -probabilities

ABSTRACTIn this article, we give an application of the theory of Monge–Ampère equations to transformation theory in probability. Before starting our analysis, we recall the definition of -characteristic function, introduced in Selvitella1 and Selvitella2 as a natural generalization of the classical characteristic function. The main novelty of this tool is that it permits to extend classical theorems, such as the law of large numbers (LLN) and the central limit theorem (CLT) to basically every distribution, upon the correct choice of a free parameter α. The motivation of this article is to show that the transformation from a pdf to its -counterpart is transferred also at the level of the random variables. We treat explicitly the cases of the multivariate normal distribution, multivariate exponential distribution, and Cauchy distribution. Moreover, we prove some rigidity theorems on the possible transformations which send a pdf to its -counterpart. In the general case, it is not possible to construct explicitly the transformation between the random variables, despite it is always possible to reduce to quadrature the transformation between the pdfs. We conclude the article by setting our theorems in the context of the Monge–Ampère equations and the optimal transportation theory and by giving some numerics to illustrate the motion of mass, while transforming a pdf to its -counterpart.

Constrained test in linear models with multivariate power exponential distribution

We investigate the problem of testing equality and inequality constraints on regression coefficients in linear models with multivariate power exponential (MPE) distribution. This distribution has received considerable attention in recent years and provides a useful generalization of the multivariate normal distribution. We examine the performance of the power of the likelihood ratio, Wald and Score tests for grouped data and in the presence of regressors, in small and moderate sample sizes, using Monte Carlo simulations. Additionally, we present a real example to illustrate the performance of the proposed tests under the MPE model.