Maia Berkane

Statistical inference based on pseudo-maximum likelihood estimators in elliptical populations

In this article we develop statistical inference based on the maximum likelihood method in elliptical populations with an unknown density function. The method assuming the multivariate normal distribution, using the sample mean and the sample covariance matrix, is basically correct even for elliptical populations under a certain kurtosis adjustment, but is not statistically efficient, especially when the kurtosis of the population distribution has higher than moderate values. On the other hand, several methods of statistical inference assuming a particular family (e.g., multivariate T distribution) of elliptical distributions have been recommended as a robust procedure against outliers or distributions with heavy tails. Such inference also will be important to maintain a high efficiency of statistical inference in elliptical populations. In practice, however, it is very difficult to choose an appropriate family of elliptical distributions, and one may misspecify the family. Furthermore, extra parameters (...

Estimation of Contamination Parameters and Identification of Outliers in Multivariate Data

Multivariate outliers may be modeled using the contaminated multivariate normal distribution with two parameters indicating the percentage of outliers and the degree of contamination. Recent developments in elliptical distribution theory are used to determine estimators of these parameters. These estimators can be used with an index of Mahalanobis distance to identify the multivariate outliers, which can then be eliminated to obtain approximately normal data. The performance of the proposed estimators and outliers rejection procedures are evaluated in a small simulation study.

Moments of elliptically distributed random variates

This paper presents an inductive method for computing the moments of an elliptically distributed random variate. A sequence of new parameters relating higher-order to second moments is introduced. The known kurtosis parameter is shown to be a member of this sequence.

Pseudo maximum likelihood estimation in elliptical theory: effects of misspecification

Abstract Recently, robust extensions of normal theory statistics have been proposed to permit modeling under a wider class of distributions (e.g., Taylor, 1992). Let X be a p × 1 random vector, μ a p × 1 location parameter, and V a p × p scatter matrix. Kano et al. (1993) studied inference in the elliptical class of distributions and gave a criterion for the choice of a particular family within this class to best describe the data at hand when the latter exhibit serious departure from normality. In this paper, we investigate the criterion for a simple but general set-up, namely, when the operating distribution is multivariate t with ν degrees of freedom and the model is also a multivariate t -distribution with α degrees of freedom. We compute the exact inefficiency of the estimators of μ and V based on that model and compare it to the one based on the mutivariate normal model. Our results provide evidence for the choice of ν = 4 proposed by Lange (1989). In addition, we give numerical results showing that for fixed ν, the inflation of the variance of the pseudo maximum likelihood estimator of the scatter matrix, as a function of the hypothesized degrees of freedom α, is increasing in its domain.

Characterizing parameters of multivariate elliptical distributions

This paper defines new parameters characterizing multivariate elliptical distributions. Mardias coefficient of multivariate kurtosis is shown to be essentially one of these parameters. A simple relation is established between centered multivariate product moments and second moments of the variables. The general results are verified on the contaminated normal distribution as an example.

Mardia's coefficient of kurtosis in elliptical populations

Mardia (1970) defined a measure of multivariate kurtosis and derived its asymptotic distribution for samples from a multivariate normal population. Some new results on elliptical distributions are used to extend Mardias results to samples from an elliptical distribution. These results provide a method for testing hypotheses on the kurtosis parameter of elliptical distributions. An appendix provides extensions of Kendall and Stuarts (1977) standard errors of bivariate moments to the third and fourth order.

Is the worst of the epidemic over? Back calculation of HIV seroprevalence in The Netherlands

This article calculates back the HIV seroprevalence in the Netherlands from AIDS cases notified 1982-1990 and rates of progression from HIV to AIDS adopted from American studies. It discusses a number of problems, such as changing AIDS definitions and the possible impact of AZT treatment. We estimate that the Netherlands had approximately 6762 HIV seropositives by the end of 1988, which is considerably lower than earlier expectations. When a hypothetical decrease of 10% in the manifestation of AIDS cases due to AZT treatment was incorporated, the estimate for the end of 1988 becomes 7549. After deduction of the AIDS patients who had died by the beginning of 1989 from this estimate, the HIV seroprevalence by the end of 1988 is approximately 7000. The distribution of seroincidence over time suggests that the HIV epidemic in our country has passed its summit and that the HIV incidence is falling quickly. The question arises as to how far this fortunate development may be considered a success of the Dutch AIDS policy, a policy characterised by more openness than in many other countries. The material studied here, however, allows no definite answer to this intriguing question.

Bias and mean square error of the maximum likelihood estimators of the parameters of the intraclass correlation model

The differential geometry of the exponential family of distributions is applied to derive the bias and the mean square error of the maximum likelihood estimator of the parameters of the intraclass correlation model.

The geometry of mean or covariance structure models in multivariate normal distributions: A unified approach

The geometric properties of mean structures of the regression model, and covariance structures of the covariance structure model, are reviewed in the light of Bates and Watts and Amaris work. Some extensions are given, namely in the context of multivariate observations for the regression model. The covariance structure model is treated in the context of the exponential family of distributions and an application to the intraclass correlation model is given. The bias of the maximum likelihood estimators of the correlation coefficient and the variance as well as the mean square error of the bias corrected estimators are derived in a closed form.