Class Of Elliptical Distributions Research Articles

An alternative stochastic model for linear portfolios

Stock market returns often tend to follow a non-normal probability distribution due to extreme losses in the tails. These cause fatter tails than normal and consequently heavy-tailed probability distributions are mostly used for modeling returns. In this work, we consider the generalized T (GT) distribution which can be heavy-tailed through its parameters and propose to use it in modeling the random stock returns. The GT distribution also contains the normal, Student t and generalized Laplace distributions as special or limiting cases of the shape parameters. The closed form expressions for the important risk measures are obtained. In case of a portfolio modeling, a multivariate extension of the distribution within the class of elliptical distributions is used. The tail mean-variance portfolio model based on the multivariate GT distribution is developed and optimal portfolio problem is solved. Risk measures for the random return of an asset whose density function is a mixture of the GT densities are obtained. The computability of all expressions derived is shown, and a real data application consisting of seven real stocks from the same sector is given.

A Cramér–Wold theorem for elliptical distributions

According to a well-known theorem of Cramér and Wold, if P and Q are two Borel probability measures on Rd whose projections PL,QL onto each line L in Rd satisfy PL=QL, then P=Q. Our main result is that, if P and Q are both elliptical distributions, then, to show that P=Q, it suffices merely to check that PL=QL for a certain set of (d2+d)/2 lines L. Moreover (d2+d)/2 is optimal. The class of elliptical distributions contains the Gaussian distributions as well as many other multivariate distributions of interest. Our theorem contrasts with other variants of the Cramér–Wold theorem, in that no assumption is made about the finiteness of moments of P and Q. We use our results to derive a statistical test for equality of elliptical distributions, and carry out a small simulation study of the test, comparing it with other tests from the literature. We also give an application to learning (binary classification), again illustrated with a small simulation.

Multivariate semiparametric control charts for mixed-type data.

A useful tool that has gained popularity in the Quality Control area is the control chart which monitors a process over time, identifies potential changes, understands variations, and eventually improves the quality and performance of the process. This article introduces a new class of multivariate semiparametric control charts for monitoring multivariate mixed-type data, which comprise both continuous and discrete random variables (rvs). Our methodology leverages ideas from clustering and Statistical Process Control to develop control charts for MIxed-type data. We propose four control chart schemes based on modified versions of the KAy-means for MIxed LArge KAMILA data clustering algorithm, where we assume that the two existing clusters represent the reference and the test sample. The charts are semiparametric, the continuous rvs follow a distribution that belongs in the class of elliptical distributions. Categorical scale rvs follow a multinomial distribution. We present the algorithmic procedures and study the characteristics of the new control charts. The performance of the proposed schemes is evaluated on the basis of the False Alarm Rate and in-control Average Run Length. Finally, we demonstrate the effectiveness and applicability of our proposed methods utilizing real-world data.

Testing identification via heteroskedasticity in structural vector autoregressive models

Summary Tests for identification through heteroskedasticity in structural vector autoregressive analysis are developed for models with two volatility states where the time point of volatility change is known. The tests are Wald-type tests for which only the unrestricted model, including the covariance matrices of the two volatility states, has to be estimated. The residuals of the model are assumed to be from the class of elliptical distributions, which includes Gaussian models. The asymptotic null distributions of the test statistics are derived, and simulations are used to explore their small-sample properties. Two empirical examples illustrate the usefulness of the tests in applied work.

Robust Semiparametric Efficient Estimators in Complex Elliptically Symmetric Distributions

Covariance matrices play a major role in statistics, signal processing and machine learning applications. This paper focuses on the semi-parametric covariance/scatter matrix estimation problem in elliptical distributions. The class of elliptical distributions can be seen as a semiparametric model where the finite-dimensional vector of interest is given by the location vector and by the (vectorized) covariance/scatter matrix, while the density generator represents an infinite-dimensional nuisance function. The main aim of this work is then to provide possible estimators of the finite-dimensional parameter vector able to reconcile the two dichotomic concepts of robustness and (semiparametric) efficiency. An R-estimator satisfying these requirements has been recently proposed by Hallin, Oja and Paindaveine for real-valued elliptical data by exploiting the Le Cam's theory of one-step efficient estimators and the rank-based statistics. In this paper, we firstly recall the building blocks underlying the derivation of such real-valued R-estimator, then its extension to complex-valued data is proposed. Moreover, through numerical simulations, its estimation performance and robustness to outliers are investigated in a finite-sample regime.

Testing Identification via Heteroskedasticity in Structural Vector Autoregressive Models

Tests for identification through heteroskedasticity in structural vector autoregressive analysis are developed for models with two volatility states where the time point of volatility change is known. The tests are Wald type tests for which only the unrestricted model including the covariance matrices of the two volatility states have to be estimated. The residuals of the model are assumed to be from the class of elliptical distributions which includes Gaussian models. The asymptotic null distributions of the test statistics are derived and simulations are used to explore their small sample properties. Two empirical examples illustrate the usefulness of the tests.

Статистические процедуры идентификации сетевых структур фондовых рынков

Network (graphical) model of stock market is a complete weighted graph. Nodes of the graph corresponds to the stocks and weights of edges are given by some measure of dependence between characteristics of the stocks. The most common characteristic of stocks is their return. In the analysis of network (graphical) models of returns of primary interest are network structures (subgraphs of a complete graph), containing key information about the considered network. Popular network structures are the minimum spanning tree, planar maximally filtered graph, market graph, a cliques and an independent set of the market graph. The problem of identification of network structure is to define the structure from observations. An important characteristic of identification statistical procedure is its uncertainty related with finite sample size. Significant role in this play the joint distribution of returns and the choice of measures of dependence between them. The most common measure of dependence is Pearson's correlation. A wide class of joint distributions of stock returns is represented by elliptical models. However, the procedures based on Pearson correlations are non robust when the joint distribution of returns is deviated from the normal in the class of elliptical distributions. The aim of this work is to present a general approach to construction of robust (distribution free) statistical procedures of identification of network structures. It is proposed to use the probability of sign coincidence of stock returns as a measure of dependence. It is shown that the single-step and stepwise standard procedure of identification of network structures based on the probability of sign coincidence are robust in the class of elliptical distributions. It allows to recommend these procedures for practical applications.

The Tail Stein's Identity with Applications to Risk Measures

In this article, we examine a generalized version of an identity made famous by Stein, who constructed the so-called Stein's Lemma in the special case of a normal distribution. Other works followed to extend the lemma to the larger class of elliptical distributions. The lemma has had many applications in statistics, finance, insurance, and actuarial science. In an attempt to broaden the application of this generalized identity, we consider the version in the case where we investigate only the tail portion of the distribution of a random variable. Understanding the tails of a distribution is very important in actuarial science and insurance. Our article therefore introduces the concept of the “tail Stein's identity” to the case of any random variable defined on an appropriate probability space with a Lebesgue density function satisfying certain regularity conditions. We also examine this “tail Stein's identity” to the class of discrete distributions. This extended identity allows us to develop recursive formulas for generating tail conditional moments. As examples and illustrations, we consider several classes of distributions including the exponential family, and we apply this result to demonstrate how to generate tail conditional moments. This holds a large promise for applications in risk management.

A characterization of elliptical distributions and some optimality properties of principal components for functional data

As in the multivariate setting, the class of elliptical distributions on separable Hilbert spaces serves as an important vehicle and reference point for the development and evaluation of robust methods in functional data analysis. In this paper, we present a simple characterization of elliptical distributions on separable Hilbert spaces, namely we show that the class of elliptical distributions in the infinite-dimensional case is equivalent to the class of scale mixtures of Gaussian distributions on the space. Using this characterization, we establish a stochastic optimality property for the principal component subspaces associated with an elliptically distributed random element, which holds even when second moments do not exist. In addition, when second moments exist, we establish an optimality property regarding unitarily invariant norms of the residuals covariance operator.

Default Probabilities and Default Correlations Under Stress

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. The analysis is performed in the class of elliptical distributions, a family of light-tailed to heavy-tailed distributions encompassing many distributions commonly found in financial modelling. It turns out that the asymptotic limit of default probabilities and default correlations depend on the max-domain of the elliptical distribution's mixing variable. In case the mixing variable is regularly varying, default probabilities are strictly smaller than 1 and default correlations are in (0, 1). Both can be expressed in terms of the Student t-distribution function. In the rapidly varying case, default probabilities are 1 and default correlations are 0. We compare our results to the tail dependence function and discuss implications for credit portfolio modelling.

A Black–Litterman asset allocation model under Elliptical distributions

In optimal portfolio allocation, Black and Litterman [Financ. Anal. J., 1992, 48, 28–43] provide for a pioneering framework of allowing to incorporate investors’ views based on a prior distribution to derive a posterior distribution of portfolio returns and optimal asset allocations. Meucci [Risk and Asset Allocation, 2005] rephrases the model in terms of investors’ views on the market, rather than just the market parameters as in the original Black and Litterman [Financ. Anal. J., 1992, 48, 28–43]. This market-based version is believed to be much more parsimonious and allows for a more natural extension to directly input views in a non-Normal market. This paper extends Meucci’s market-based version of the Black–Litterman model to the case when returns in the market fall within the class of Elliptical distributions, while also importantly preserving the equilibrium-based assumption in the model. Here, within this class for which the Normal distribution is a special case, we develop the explicit form of the posterior distribution after considering proper conditional conjugate-type prior distributions. This resulting posterior allows us to obtain solutions to optimization problems of asset allocation based on a variety of risk measures (e.g. Mean-Variance, Mean-VaR, Mean-Conditional VaR). Elliptical models of portfolio returns have recently crept into the financial literature because of their greater flexibility to accommodate larger tails. As numerical demonstrations, we examine how these principles work in a portfolio with international stock indices and show why models with more flexible tails can affect the asset allocation.

Modified likelihood ratio tests in heteroskedastic multivariate regression models with measurement error

In this paper, we develop modified versions of the likelihood ratio test for multivariate heteroskedastic errors-in-variables regression models. The error terms are allowed to follow a multivariate distribution in the elliptical class of distributions, which has the normal distribution as a special case. We derive the Skovgaard-adjusted likelihood ratio statistics, which follow a chi-squared distribution with a high degree of accuracy. We conduct a simulation study and show that the proposed tests display superior finite sample behaviour as compared to the standard likelihood ratio test. We illustrate the usefulness of our results in applied settings using a data set from the WHO MONICA Project on cardiovascular disease.

On the Realized Risk of High-Dimensional Markowitz Portfolios

We study the realized risk of Markowitz portfolios computed using parameters estimated from data and generalizations to similar questions involving the out-of-sample risk in quadratic programs with linear equality constraints. We do so under the assumption that the data is generated according to an elliptical model, which allows us to study models with heavy tails, tail dependence, and leptokurtic marginals for the data. We place ourselves in the setting of high-dimensional inference where the number of assets in the portfolio, $p$, is large and comparable to the number of samples, $n$, we use to estimate the parameters. Our approach is based on random matrix theory. We consider the impact of both the estimation of the mean and of the covariance. Our work shows that risk is underestimated in this setting and, further, that in the class of elliptical distributions, the Gaussian case yields the least amount of risk underestimation. The problem is more pronounced for genuinely elliptical distributions. There...

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.

A multivariate ultrastructural errors-in-variables model with equation error

This paper deals with asymptotic results on a multivariate ultrastructural errors-in-variables regression model with equation errors. Sufficient conditions for attaining consistent estimators for model parameters are presented. Asymptotic distributions for the line regression estimators are derived. Applications to the elliptical class of distributions with two error assumptions are presented. The model generalizes previous results aimed at univariate scenarios.

A Black-Litterman Asset Allocation Model Under Elliptical Distributions

In optimal portfolio allocation, Black and Litterman (1992) provide for a pioneering framework of allowing to incorporate investors’ views based on a prior distribution to derive a posterior distribution of portfolio returns and optimal asset allocations. Meucci (2005) rephrases the model in terms of investors’ views on the market, rather than just the market parameters as in the original Black and Litterman (1992). This market-based version is believed to be much more parsimonious and allows for a natural extension to directly input views in a non-Normal market. This paper extends Meucci’s market-based version of the Black-Litterman model to the case when returns in the market fall within the class of Elliptical distributions, while also importantly preserving the equilibrium-based assumption in the model. Here within this class for which the Normal distribution is a special case, we develop the explicit form of the posterior distribution after considering proper conditional conjugate-type prior distributions. This resulting posterior allows us to obtain solutions to optimization problems of asset allocation based on a variety of risk measures (e.g. Mean-Variance, Mean-VaR, Mean-Conditional VaR). Elliptical models of portfolio returns have recently crept into the financial literature because of its greater flexibility to accommodate larger tails. As a numerical demonstration, we examine how these principles work in a portfolio with international stock indices.

Statistical analysis of proficiency testing results under elliptical distributions

The use of inter-laboratory test comparisons to determine the performance of individual laboratories for specific tests (or for calibration) [ISO/IEC Guide 43-1, 1997. Proficiency testing by interlaboratory comparisons — Part I: Development and operation of proficiency testing schemes] is called Proficiency Testing (PT). In this paper we propose the use of the generalized likelihood ratio test to compare the performance of the group of laboratories for specific tests relative to the assigned value and illustrate the procedure considering an actual data from the PT program in the area of volume. The proposed test extends the test criteria in use allowing to test for the consistency of the group of laboratories. Moreover, the class of elliptical distributions are considered for the obtained measurements.

Contaminated Normal Type Symmetry Model and Decomposition of Symmetry for Square Contingency Tables

For the analysis of square contingency tables with ordered categories, Agresti (1983) considered the linear diagonals-parameter symmetry (LDPS) model being an extension of the symmetry (S) model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. This paper proposes another extension of the S model which may be appropriate for a square ordinal table if it is reasonable to assume an underlying e-contaminated bivariate normal distribution with equal marginal variances included in the class of elliptical distributions. The proposed model has a structure of weighted mixture of two LDPS models. Moreover, this paper gives a theorem that the S model holds if and only if both the proposed model and the marginal means equality model hold. The simulation studies based on the e-contaminated bivariate normal distribution are given.

An extension of Wick’s theorem

We propose an extension of a result by Repetowicz et al. [Repetowicz, P., Richmond, P., 2004. The Wick theorem for non-Gaussian distributions and its application for noise filtering of correlated q -exponentially distributed random variables (in press). arxiv:math-ph/0411020 v1] about Wick’s theorem and its applications. We first show that Wick’s theorem can be extended to the uniform distribution on the sphere and then to the whole class of elliptical distributions. Then, as a special case, we detail this result for distributions that are scale mixtures of Gaussians. Finally, we show that these results allow to recover easily a theorem by Folland [Folland, G.B., 2001. How to integrate a polynomial over a sphere. The American Mathematical Monthly 108 (5), 446–448] about the integration of polynomials over the sphere.

On the asymptotic distributions of two statistics for two-level covariance structure models within the class of elliptical distributions

Since data in social and behavioral sciences are often hierarchically organized, special statistical procedures for covariance structure models have been developed to reflect such hierarchical structures. Most of these developments are based on a multivariate normality distribution assumption, which may not be realistic for practical data. It is of interest to know whether normal theory-based inference can still be valid with violations of the distribution condition. Various interesting results have been obtained for conventional covariance structure analysis based on the class of elliptical distributions. This paper shows that similar results still hold for 2-level covariance structure models. Specifically, when both the level-1 (within cluster) and level-2 (between cluster) random components follow the same elliptical distribution, the rescaled statistic recently developed by Yuan and Bentler asymptotically follows a chi-square distribution. When level-1 and level-2 have different elliptical distributions, an additional rescaled statistic can be constructed that also asymptotically follows a chi-square distribution. Our results provide a rationale for applying these rescaled statistics to general non-normal distributions, and also provide insight into issues related to level-1 and level-2 sample sizes.