Multivariate Elliptical Distribution Research Articles

Multisensor Estimation Fusion on Statistical Manifold.

In the paper, we characterize local estimates from multiple distributed sensors as posterior probability densities, which are assumed to belong to a common parametric family. Adopting the information-geometric viewpoint, we consider such family as a Riemannian manifold endowed with the Fisher metric, and then formulate the fused density as an informative barycenter through minimizing the sum of its geodesic distances to all local posterior densities. Under the assumption of multivariate elliptical distribution (MED), two fusion methods are developed by using the minimal Manhattan distance instead of the geodesic distance on the manifold of MEDs, which both have the same mean estimation fusion, but different covariance estimation fusions. One obtains the fused covariance estimate by a robust fixed point iterative algorithm with theoretical convergence, and the other provides an explicit expression for the fused covariance estimate. At different heavy-tailed levels, the fusion results of two local estimates for a static target display that the two methods achieve a better approximate of the informative barycenter than some existing fusion methods. An application to distributed estimation fusion for dynamic systems with heavy-tailed process and observation noises is provided to demonstrate the performance of the two proposed fusion algorithms.

Semiparametric estimation of the high-dimensional elliptical distribution

This paper investigates semiparametric estimation of the multivariate elliptical distribution in case dimensionality increases with the sample size. We prove the almost sure convergence and derive the convergence rates of the estimator, depending on the sample size, dimensionality, and the bandwidth of the kernel. As an important by-product, we show almost sure convergence with the corresponding convergence rates for the sample covariance matrix under the Frobenius norm. An extensive simulation study has supported the theory.

A broad class of multivariate distributions for rates and proportions

This paper introduces a general class of multivariate distributions for rates and proportions on the basis of the multivariate elliptical distribution. We derive various general properties of this multivariate distribution. The model parameters are estimated using the maximum likelihood estimation method. In particular, we have considered parameter estimation in detail for two special cases of this multivariate family, using multivariate normal kernel and multivariate Student-t kernel. Hypothesis testing inference for the correlation matrix on the basis of the likelihood ratio test statistic is also proposed. Empirical applications that employ real data are considered to illustrate the usefulness of the new class of multivariate distributions for rates and proportions in practice.

Revitalizing the multivariate elliptical leptokurtic-normal distribution and its application in model-based clustering

We derive an improved unimodality criteria for the elliptical multivariate leptokurtic-normal (MLN) distribution. For finite mixtures of MLN, we prove identifiability. Then we provide a new estimation algorithm and show improvement over existing methods.

Efficient Simulation for Expectations over the Union of Half-Spaces

We consider the problem of estimating expectations over the union of half-spaces. Such a problem arises in many applications such as option pricing and stochastic activity networks. More recent applications include systemic risk measurements of financial networks. Assuming that random variables follow a multivariate elliptical distribution, we develop a conditional Monte Carlo method and prove its asymptotic efficiencies. We then demonstrate the numerical performance of the proposed method in three different application areas.

Improved estimation in a general multivariate elliptical model

The problem of reducing the bias of maximum likelihood estimator in a general multivariate elliptical regression model is considered. The model is very flexible and allows the mean vector and the dispersion matrix to have parameters in common. Many frequently used models are special cases of this general formulation, namely: errors-in-variables models, nonlinear mixed-effects models, heteroscedastic nonlinear models, among others. In any of these models, the vector of the errors may have any multivariate elliptical distribution. We obtain the second-order bias of the maximum likelihood estimator, a bias-corrected estimator, and a bias-reduced estimator. Simulation results indicate the effectiveness of the bias correction and bias reduction schemes.

Multivariate moment based extreme value index estimators

Modeling extreme events is of paramount importance in various areas ofscience — biostatistics, climatology, finance, geology, and telecommunications, toname a few. Most of these application areas involve multivariate data. Estimationof the extreme value index plays a crucial role in modeling rare events. There isan affine invariant multivariate generalization of the well known Hill estimator—theseparating Hill estimator. However, the Hill estimator is only suitable for heavy taileddistributions. As in the case of the separating multivariate Hill estimator, we considerestimation of the extreme value index under the assumption of multivariate ellipticity.We provide affine invariant multivariate generalizations of the moment estimator andthe mixed moment estimator. These estimators are suitable for both: light and heavytailed distributions. Asymptotic properties of the new extreme value index estimatorsare derived under multivariate elliptical distribution with known location and scatter.The effect of replacing true location and scatter by estimates is examined in a thoroughsimulation study.

Multivariate Moment Based Extreme Value Index Estimators

Modeling extreme events is of paramount importance in various areas of science — biostatistics, climatology, finance, geology, and telecommunications, to name a few. Most of these application areas involve multivariate data. Estimation of the extreme value index plays a crucial role in modeling rare events. There is an affine invariant multivariate generalization of the well known Hill estimator — the separating Hill estimator. However, the Hill estimator is only suitable for heavy tailed distributions. As in the case of the separating multivariate Hill estimator, we consider estimation of the extreme value index under the assumption of multivariate ellipticity. We provide affine invariant multivariate generalizations of the moment estimator and the mixed moment estimator. These estimators are suitable for both: light and heavy tailed distributions. Asymptotic properties of the new extreme value index estimators are derived under multivariate elliptical distribution with known location and scatter. The effect of replacing true location and scatter by estimates is examined in a thorough simulation study.

On Asymptotic Properties of the Separating Hill Estimator

Modeling and understanding multivariate extreme events is challenging, but of great importance in various applications - e.g. in biostatistics, climatology, and finance. The separating Hill estimator can be used in estimating the extreme value index of a heavy tailed multivariate elliptical distribution. We consider the asymptotic behavior of the separating Hill estimator under estimated location and scatter. The asymptotic properties of the separating Hill estimator are known under elliptical distribution with known location and scatter. However, the effect of estimation of the location and scatter has previously been examined only in a simulation study. We show, analytically, that the separating Hill estimator is consistent and asymptotically normal under estimated location and scatter, when certain mild conditions are met.

On the properties of the constrained Hansen–Jagannathan distance

We provide an in-depth analysis of the theoretical properties of the Hansen–Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. Under a multivariate elliptical distribution assumption, we present explicit expressions for the HJ-distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the stochastic discount factor (SDF) parameters in the case of linear SDFs. This allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to evaluate and rank models. We also study the asymptotic and finite-sample properties of the sample constrained HJ-distance. Finally, we demonstrate the practical relevance of our theoretical findings in an empirical illustration of some popular asset-pricing models.

Systematic Scenario Selection: A Methodology for Selecting a Representative Grid of Shock Scenarios from a Multivariate Elliptical Distribution

We present a technique for selecting multidimensional shock scenarios for use in financial stress testing. The methodology systematically enforces internal consistency among the shock dimensions by sampling points of arbitrary severity from a plausible joint probability distribution. The approach involves a grid search of sparse, well distributed, stress-test scenarios, which we regard as a middle ground between traditional stress testing and reverse stress testing. Choosing scenarios in this way reduces the danger of blind spots in stress testing. We suggest extensions to address the issues of non-monotonic loss functions and univariate shocks. We provide tested and commented source code in Matlab®.

Search before Trade-Offs are Known

Search, broadly defined, is a critical managerial activity. Our contribution is a model of search for multiattribute alternatives. Most of the search literature considers univariate alternatives, and can be applied to a multiattribute setting provided the trade-offs to be used at the final selection stage were known at the search stage. However, uncertainty about tradeoffs is likely to occur, especially in settings (e.g., vendor selection, new product development, innovation tournaments) that involve parallel search. We show that incorporating uncertainty about trade-offs into a model changes its search strategy recommendations. Failing to account for such uncertainty, which is likely in practice, leads to suboptimal search and potentially large losses. For parallel search and a multivariate elliptical (e.g., normal) distribution of the alternatives, the solution is equivalent to univariate search with appropriately adjusted standard deviation. We prove that the optimal number of draws there increases if uncertainty about tradeoffs increases.

Search Before Trade-offs Are Known

Search, broadly defined, is a critical managerial activity. Our contribution is a model of search for multiattribute alternatives, and our focus is on parallel search, where the decision is about the number of alternatives to explore. Most of the search literature considers univariate alternatives, and it can be applied to a multiattribute setting provided that the trade-offs to be used at the final selection stage were known at the search stage. However, uncertainty about trade-offs is likely to occur, especially in settings that involve parallel search (e.g., vendor selection, new product development, innovation tournaments). We show that incorporating uncertainty about trade-offs into a model changes its search strategy recommendations. Failing to account for such uncertainty, which is likely in practice, leads to suboptimal search and potentially large losses. For parallel search and a multivariate elliptical (e.g., normal) distribution of the alternatives, the solution is equivalent to univariate search with appropriately adjusted standard deviation. We prove that, in this setting, the optimal number of alternatives to explore increases if uncertainty about trade-offs increases, and we discuss the value of information about uncertain trade-offs.

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X(⁎)AX, when A=A(⁎) is a positive semidefinite matrix, X has a matrix multivariate elliptical distribution and X(⁎) denotes the usual conjugate transpose of X. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.

A characterization of optimal portfolios under the tail mean–variance criterion

The tail mean–variance model was recently introduced for use in risk management and portfolio choice; it involves a criterion that focuses on the risk of rare but large losses, which is particularly important when losses have heavy-tailed distributions. If returns or losses follow a multivariate elliptical distribution, the use of risk measures that satisfy certain well-known properties is equivalent to risk management in the classical mean–variance framework. The tail mean–variance criterion does not satisfy these properties, however, and the precise optimal solution typically requires the use of numerical methods. We use a convex optimization method and a mean–variance characterization to find an explicit and easily implementable solution for the tail mean–variance model. When a risk-free asset is available, the optimal portfolio is altered in a way that differs from the classical mean–variance setting. A complete solution to the optimal portfolio in the presence of a risk-free asset is also provided.

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X0, XT, YT, ZT)T having a multivariate elliptical distribution, we derive the exact joint distribution of (X0, aTX(n), bTY[n], cTZ[n])T, where a, b, c∈ℝn, X(n)=(X(1), …, X(n))T, X(1)<···<X(n), is the vector of order statistics arising from X, and Y[n]=(Y[1], …, Y[n])T and Z[n]=(Z[1], …, Z[n])T denote the vectors of concomitants corresponding to X(n) ((Y[r], Z[r])T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X0, X(r), Y[r], Z[r])T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X(r), say, based on the concomitants Y[r] and Z[r]. Finally, we illustrate the usefulness of our results by a real data.

Analytical Solution for the Constrained Hansen-Jagannathan Distance Under Multivariate Ellipticity

We provide an in-depth analysis of the theoretical properties of the Hansen-Jagannathan (HJ) distance that incorporates a no-arbitrage constraint. Under a multivariate elliptical distribution assumption, we present explicit expressions for the HJ-distance with a no-arbitrage constraint, the associated Lagrange multipliers, and the SDF parameters in the case of linear SDFs. This approach allows us to analyze the benefits and costs of using the HJ-distance with a no-arbitrage constraint to rank asset pricing models.

Multi-stock portfolio optimization under prospect theory

We study how a behavioral agent allocates her portfolio. We consider a cumulative prospect theory investor in a single period setting with one riskless bond and multiple risky stocks, which follow a multivariate elliptical distribution. Our main result is a two-fund separation between the riskless bond and a mean–variance-portfolio, up to an exogenous benchmark portfolio. The mean–variance-portfolio, which we derive explicitly, is the same for all agents. Individual risk preferences are mirrored only in the participation in this portfolio. This dependence is illustrated by considering empirical returns. Furthermore we solve ill-posed optimization problems by imposing a regulatory risk constraint. Finally we address specific parameterizations of the value function by studying power, linear, and exponential utility.

On Wishart distribution: Some extensions

This paper proposes a unified approach that enables the Wishart distribution to be studied simultaneously in the real, complex, quaternion and octonion cases under elliptical models. In particular, the matrix multivariate elliptical distribution, the noncentral generalised Wishart distribution, the joint density of the eigenvalues and the distribution of the maximum eigenvalue are obtained for real normed division algebras.

On the distribution of the sample autocorrelation coefficients

Sample autocorrelation coefficients are widely used to test the randomness of a time series. Despite its unsatisfactory performance, the asymptotic normal distribution is often used to approximate the distribution of the sample autocorrelation coefficients. This is mainly due to the lack of an efficient approach in obtaining the exact distribution of sample autocorrelation coefficients. In this paper, we provide an efficient algorithm for evaluating the exact distribution of the sample autocorrelation coefficients. Under the multivariate elliptical distribution assumption, the exact distribution as well as exact moments and joint moments of sample autocorrelation coefficients are presented. In addition, the exact mean and variance of various autocorrelation-based tests are provided. Actual size properties of the Box–Pierce and Ljung–Box tests are investigated, and they are shown to be poor when the number of lags is moderately large relative to the sample size. Using the exact mean and variance of the Box–Pierce test statistic, we propose an adjusted Box–Pierce test that has a far superior size property than the traditional Box–Pierce and Ljung–Box tests.