Concomitants of multivariate order statistics from multivariate elliptical distributions

ABSTRACT

In this article, we consider a (k + 1)n-dimensional elliptically contoured random vector (X^T₁, X₂^T, …, X^T_k, Z^T)^T = (X₁₁, …, X_1n, …, X_k1, …, X_kn, Z₁, …, Z_n)^T and derive the distribution of concomitant of multivariate order statistics arising from X₁, X₂, …, X_k. Specially, we derive a mixture representation for concomitant of bivariate order statistics. The joint distribution of the concomitant of bivariate order statistics is also obtained. Finally, the usefulness of our result is illustrated by a real-life data.     

MSE performance and minimax regret significance points for a HPT estimator under a multivariate t error distribution

In this article, assuming that the error terms follow a multivariate t distribution,we derive the exact formulae forthe moments of the heterogeneous preliminary test (HPT) estimator proposed by Xu (2012b Xu, H. (2012b). MSE performance and minimax regret significance points for a HPT estimator when each individual regression coefficient is estimated. Commun. Stat. Theory Methods 42:2152–2164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We also execute the numerical evaluation to investigate the mean squared error (MSE) performance of the HPT estimator and compare it with those of the feasible ridge regression (FRR) estimator and the usual ordinary least squared (OLS) estimator. Further, we derive the optimal critical values of the preliminary F test for the HPT estimator, using the minimax regret function proposed by Sawa and Hiromatsu (1973 Sawa, T., Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41:1093–1101.[Crossref], [Web of Science ®] , [Google Scholar]). Our results show that (1) the optimal significance level (α*) increases as the degrees of freedom of multivariate t distribution (ν₀) increases; (2) when ν₀ ? 10, the value of α* is close to that in the normal error case.     

A conditional approach for multivariate extreme values (with discussion)

Summary.  Multivariate extreme value theory and methods concern the characterization, estimation and extrapolation of the joint tail of the distribution of a d -dimensional random variable. Existing approaches are based on limiting arguments in which all components of the variable become large at the same rate. This limit approach is inappropriate when the extreme values of all the variables are unlikely to occur together or when interest is in regions of the support of the joint distribution where only a subset of components is extreme. In practice this restricts existing methods to applications where d is typically 2 or 3. Under an assumption about the asymptotic form of the joint distribution of a d -dimensional random variable conditional on its having an extreme component, we develop an entirely new semiparametric approach which overcomes these existing restrictions and can be applied to problems of any dimension. We demonstrate the performance of our approach and its advantages over existing methods by using theoretical examples and simulation studies. The approach is used to analyse air pollution data and reveals complex extremal dependence behaviour that is consistent with scientific understanding of the process. We find that the dependence structure exhibits marked seasonality, with ex- tremal dependence between some pollutants being significantly greater than the dependence at non-extreme levels.     

Concomitants of order statistics from multivariate elliptical distributions

In this paper, by considering a 2n-dimensional elliptically contoured random vector (X^T,Y^T)^T=(X₁,…,X_n,Y₁,…,Y_n)^T, we derive the exact joint distribution of linear combinations of concomitants of order statistics arising from X. Specifically, we establish a mixture representation for the distribution of the rth concomitant order statistic, and also for the joint distribution of the rth order statistic and its concomitant. We show that these distributions are indeed mixtures of multivariate unified skew-elliptical distributions. The two most important special cases of multivariate normal and multivariate t distributions are then discussed in detail. Finally, an application of the established results in an inferential problem is outlined.     

A semi-parametric model for multivariate extreme values

Threshold methods for multivariate extreme values are based on the use of asymptotically justified approximations of both the marginal distributions and the dependence structure in the joint tail. Models derived from these approximations are fitted to a region of the observed joint tail which is determined by suitably chosen high thresholds. A drawback of the existing methods is the necessity for the same thresholds to be taken for the convergence of both marginal and dependence aspects, which can result in inefficient estimation. In this paper an extension of the existing models, which removes this constraint, is proposed. The resulting model is semi-parametric and requires computationally intensive techniques for likelihood evaluation. The methods are illustrated using a coastal engineering application.     

Fragility index of block tailed vectors

Financial crises are a recurrent phenomenon with important effects on the real economy. The financial system is inherently fragile and it is therefore of great importance to be able to measure and characterize its systemic stability. Multivariate extreme value theory provide us such a framework through the fragility index ( 11, 7 and 8). Here we generalize this concept and contribute to the modeling of the stability of a stochastic system divided into blocks. We will find several relations with well-known tail dependence measures in the literature, which will provide us immediate estimators. We end with an application to financial data.     

An estimator for the extreme-value index

Asymptotic normally is derived for a new estimator of the extreme value index under certain conditions.     

A pseudo restricted MLE under multivariate order restrictions and its algorithm

Abstract

It is shown in this paper that a quasi order for the vectors in R^p is a cone induced if and only if the order is preservable under limits and under linear combinations with non-negative coefficients. For the mean vectors in MANOVA subject to the restriction of simple ordering, a pseudo restricted MLE is proposed. This estimator is a matrix projection onto a closed convex set inside the restricted domain. An algorithm for the pseudo restricted MLE is developed, that computes the matrix projections using only vector projections.     

On an adaptive transformation–retransformation estimate of multivariate location

An affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied. The work is primarily motivated by earlier work on different versions of the multivariate median and their properties. We explore an issue related to efficiency and equivariance that was originally raised by Bickel and subsequently investigated by Brown and Hettmansperger. Our estimate has better asymptotic performance than the vector of co-ordinatewise medians when the variables are substantially correlated. The finite sample performance of the estimate is investigated by using Monte Carlo simulations. Some examples are presented to demonstrate the effect of the adaptive transformation–retransformation strategy in the construction of multivariate location estimates for real data.     

Bivariate Tail Dependence and the Generation of Multivariate Extreme Value Distributions

We define, in a probabilistic way, a parametric family of multivariate extreme value distributions. We derive its copula, which is a mixture of several complete dependent copulas and total independent copulas, and the bivariate tail dependence and extremal coefficients. Based on the obtained results for these coefficients, we propose a method to build multivariate extreme value distributions with prescribed tail/extremal coefficients. We illustrate the results with examples.     

On the estimation of extreme directional multivariate quantiles

Abstract

In multivariate extreme value theory (MEVT), the focus is on analysis outside of the observable sampling zone, which implies that the region of interest is associated to high risk levels. This work provides tools to include directional notions into the MEVT, giving the opportunity to characterize the recently introduced directional multivariate quantiles (DMQ) at high levels. Then, an out-sample estimation method for these quantiles is given. A bootstrap procedure carries out the estimation of the tuning parameter in this multivariate framework and helps with the estimation of the DMQ. Asymptotic normality for the proposed estimator is provided and the methodology is illustrated with simulated data-sets. Finally, a real-life application to a financial case is also performed.     

Abelian and Tauberian Theorems on the Bias of the Hill Estimator

The bias of Hill's estimator for the positive extreme value index of a distribution is investigated in relation to the convergence rate in the regular variation property of the tail function of the common distribution of the sample and the corresponding tail quantile function. Based on the theory of generalized regular variation, natural second-order conditions are proposed which both imply and are implied by convergence of the expectation of Hill's estimator to the extreme value index at certain rates. A comparison with second-order conditions encountered in the literature is made.     

An L1-type estimator of multivariate location and shape

The main goal of the paper is to specify a suitable multivariate multilevel model for polytomous responses with a non-ignorable missing data mechanism in order to determine the factors which influence the way of acquisition of the skills of the graduates and to evaluate the degree programmes on the basis of the adequacy of the skills they give to their graduates. The application is based on data gathered by a telephone survey conducted, about two years after the degree, on the graduates of year 2000 of the University of Florence. A multilevel multinomial logit model for the response of interest is fitted simultaneously with a multilevel logit model for the selection mechanism by means of maximum likelihood with adaptive Gaussian quadrature. In the application the multilevel structure has a crucial role, while selection bias results negligible. The analysis of the empirical Bayes residuals allows to detect some extreme degree programmes to be further inspected.     

A note on Stein's lemma for multivariate elliptical distributions

When two random variables are bivariate normally distributed Stein's original lemma allows to conveniently express the covariance of the first variable with a function of the second. Landsman and Neslehova (2008) extend this seminal result to the family of multivariate elliptical distributions. In this paper we use the technique of conditioning to provide a more elegant proof for their result. In doing so, we also present a new proof for the classical linear regression result that holds for the elliptical family.     

On a multivariate log-gamma distribution and the use of the distribution in the Bayesian analysis

In this article, we propose a new generalized multivariate log-gamma distribution. We consider the usage of the proposed multivariate distribution as the prior distribution in the Bayesian analysis. The generalized multivariate log-gamma distribution allows for the inclusion of prior knowledge on correlations between model parameters when likelihood is not in the form of a normal distribution. Use of the proposed distribution in the Bayesian analysis of log-linear models is also discussed.     

On the multivariate predictive distribution of multi-dimensional effective dose: a Bayesian approach

We propose a Bayesian procedure to sample from the distribution of the multi-dimensional effective dose. This effective dose is the set of dose levels of multiple predictive factors that produce a binary response with a fixed probability.We apply our algorithms to parametric and semiparametric logistics regression models, respectively. The graphical display of random samples obtained through Markov chain Monte Carlo can provide some insight into the predictive distribution.     

On the minimaxiy of the maximum likelihood estimator in a multivariate problem

This study looks at the minimaxity of the maximum likelihood estimator (m.1.e), of the mean of a p-normal population, that has been given by Dahel, Giri and Lepage (1985). This estimator is computed on the basis of three independent samples: the first one is drawn from the whole vector of dimension p and the two others are based on the first p₁ and the last p₂ components respectively, such as p₁ +p₂=p.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.