/pdf/statistical-inference-in-elliptically-contoured-and-related-4awu8rc4oa.pdf

Statistical inference in elliptically contoured and related distributions

This article examines the adjustment of normal theory methods for the analysis of covariance structures to make them applicable under the class of elliptical distributions. It is shown that if the model satisfies a mild scale invariance condition and the data have an elliptical distribution, the asymptotic covariance matrix of sample covariances has a structure that results in the retention of many of the asymptotic properties of normal theory methods. If a scale adjustment is applied, the likelihood ratio tests of fit have the usual asymptotic chi-squared distributions. Difference tests retain their property of asymptotic independence, and maximum likelihood estimators retain their relative asymptotic efficiency within the class of estimators based on the sample covariance matrix. An adjustment to the asymptotic covariance matrix of normal theory maximum likelihood estimators for elliptical distributions is provided. This adjustment is particularly simple in models for patterned covariance or co...

Analysis of Covariance Structures under Elliptical Distributions

In this paper, a method for multivariate testing based on low-dimensional, data-dependent, linear scores is proposed. The new approach reduces the dimensionality of observations and increases the stability of the solutions. The method is reliable, even if there are many redundant variables. As a key feature, the score coefficients are chosen such that a left-spherical distribution of the scores is reached under the null hypothesis. Therefore, well-known tests become applicable in high-dimensional situations, too. The presented strategy is an alternative to least squares and maximum likelihood approaches. In a natural way, standard problems of multivariate analysis thus induce the occurrence of left-spherical, nonnormal distributions. Hence, new fields of application are opened up to the generalized multivariate analysis. The proposed methodology is not restricted to normally distributed data, but can also be extended to any left-spherically distributed observations.

/pdf/multivariate-tests-based-on-left-spherically-distributed-4c5od9pau4.pdf

Multivariate tests based on left-spherically distributed linear scores

The benchmark theory of mathematical finance is the Black–Scholes–Merton (BSM) theory, based on Brownian motion as the driving noise process for stock prices. Here the distributions of financial returns of the stocks in a portfolio are multivariate normal. Risk management based on BSM underestimates tails. Hence estimation of tail behaviour is often based on extreme value theory (EVT). Here we discuss a semi-parametric replacement for the multivariate normal involving normal variance–mean mixtures. This allows a more accurate modelling of tails, together with various degrees of tail dependence, while (unlike EVT) the whole return distribution can be modelled. We use a parametric component, incorporating the mean vector μ and covariance matrix Σ, and a non-parametric component, which we can think of as a density on [0,∞), modelling the shape (in particular the tail decay) of the distribution. We work mainly within the family of elliptically contoured distributions, focusing particularly on normal ...

A semi-parametric approach to risk management

In this paper we first show how the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity, and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of independent Beta random variables. Then, we show how near-exact distributions for the logarithms of these statistics may be expressed as Generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n−j)/n for j=2,…,p, where for three of the statistics, p is the number of variables involved, while for the other one, it is the sum of the number of variables involved and the number of mean vectors being tested. What is interesting is that the similarities exhibited by these statistics are even more striking in terms of near-exact distributions than in terms of exact distributions. Moreover all the l.r.t. statistics that may be built as products of these basic statistics also inherit a similar structure for their near-exact distributions. To illustrate this fact, an application is made to the l.r.t. statistic to test the equality of several multivariate Normal distributions.

A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis

For a class of multivariate elliptically contoured distributions the maximum-likelihood estimators of the mean vector and covariance matrix are found under certain conditions. Likelihood-ratio criteria are obtained for a class of null hypotheses. These have the same form as in the normal case.

Huang Hsu

Papers

Maximum-likelihood estimates and likelihood-ratio criteria for multivariate elliptically contoured distributions