Resumen de Extremality in multivariate statistics

Henry Laniado Rodas

• Multivariate order is a valuable tool for analyzing data properties and for extending univariate concepts based on order such as median, range, extremes, quantiles or order statistics to multivariate data. Generalizing such concepts to the multivariate case is not straightforward. While different ways of generalizing quantiles have been studied by Chaudhuri [10], a description of extensions of concepts such a median, range and quantiles to the multivariate framework has been provided by Barnett [3]. The key problem, however, in generalizing these concepts to several dimensions is the lack of a unique criterion for ordering multivariate observations. Over the last few decades, multivariate stochastic orders have also become a powerful means of comparing random vectors, especially in situations where the distributions are partially known. In particular, multivariate stochastic orders have a wide range of applications in portfolio theory. The thesis is motivated by the aspects mentioned above and its purpose is threefold. Firstly it introduces the multivariate extremality as a methodology that measures the farness of a point x with respect to a data cloud or a distribution function. We study the main properties of this new concept, as well as asymptotic results, and define a new multivariate data order based on rotations. This data order allows us to introduce a new version of the multivariate quantile which can be seen as a generalization of definitions previously studied in the literature. As a consequence of this ordering, we are able to develop an application in finance by defining a new version of the multivariate Value at Risk. Secondly, we develop a new multivariate stochastic order based on directions for generalizing the well- known orthant orders. Some examples are presented of applications in portfolio comparisons. Particular attention is paid to applications in which the use of directions is well justified in determining optimal allocations of wealth among risks in single period portfolio problems. Thirdly, the thesis aims to investigate an alternative methodology for selecting the portfolio weights in a data set that represents returns of n assets for investing. We also define new concepts of efficient frontiers based on the initial idea of Markowitz. We apply the extremality multivariate data order to order feasible portfolios in a direction that depends on specific indexes; these may be chosen by the investor and may be different from the classical variance-return in Markowitz’s model. The thesis is organized as follows: in Chapter 1 we provide a brief review of some multivariate data orders introduced in the literature in order to extend univariate statistical concepts to the multivariate setting. Following some multivariate stochastic orderings, we examine different means of comparing random vectors based on their survival and distribution functions. Finally, the chapter presents a brief introduction to the portfolio selection problem. In Chapter 2, we propose a new approach for analyzing multivariate extremes. It provides a multivariate data order based on a concept that we will call “extremality”. We establish the most relevant properties of this extremality measure and we give the theoretical basis for its nonparametric estimation. Finally, we include an application in Finance, we define an oriented multivariate Value at Risk (VaR) with level sets built through extremality which is computationally feasible in high dimensions. The results of Chapter 3 concern a new multivariate stochastic order that compares random vectors in a direction which is determined by a unit vector, generalizing the well-known upper and lower orthant orders. The main properties of this new order, together with its relationships with other multivariate stochastic orders are investigated. We also present some examples of application in the determination of optimal allocations of wealth among risks in single period portfolio problems. In Chapter 4, we introduce new concepts of efficient frontier that depend on some indexes that may be chosen by the investor and that are different from the classical variance- return in Markowitz’s model. Feasible portfolios are built with MonteCarlo simulations and the new efficient frontiers are estimated by using an extremality order to sort portfolios. The performance of the selection method is illustrated with real data. Finally, in Chapter 5, we present some general conclusions and summarize the main contributions of the thesis. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------