Study of multivariate asymmetric distributions using univariate two-piece distributions

Abstract

Classical symmetric distributions like the normal distribution are widely used. However, in reality data often display a lack of symmetry. Multiple distributions have been developed to specifically cope with asymmetric data. These can be grouped under the name “skewed distributions”. The most well known families of skewed distributions are the skew-symmetric family (univariate) or skewelliptical family (multivariate), with seminal contributions by Adelchi Azzalini. A second large family consists of two-piece distributions, though these usually appear in a univariate context. A first part of this thesis consists of the study of two families of multivariate skewed distributions constructed using univariate asymmetric two-piece distributions. For both families the focus lies on statistical inference. The first family, studied in Chapter 2, is derived by taking affine combinations of independent univariate skewed distributions. These univariate skewed distributions are members of a flexible family of asymmetric distributions and are an important basis for achieving statistical inference. Besides basic properties of the proposed distributions, also statistical inference based on a maximum likelihood approach is presented. Under some mild conditions, weak consistency and asymptotic normality of the maximum likelihood estimators are shown to hold. These results are backed up by a simulation study which confirms the developed theoretical results and some real data examples to illustrate practical applicability. The second family of multivariate skewed distributions studied, consists of incorporating flexible univariate skewed distributions into a copula structure. With an increased awareness for possible asymmetry in data, skewed copulas in combination with classical marginals have been employed to appropriately model these data. The reverse, skewed marginals with a (classical) copula, has also been considered, but mainly with skew-symmetrical marginals. In the third chapter we rely on a large family of asymmetric two-piece distributions for the univariate marginal distributions. Together with any copula, this family of asymmetric univariate distributions provides a powerful tool for skewed multivariate distributions. We discuss maximum likelihood estimation of all parameters involved. A key step in achieving statistical inference results is an extension of the theory available for generalized method of moments, under non-standard conditions. This together with the inference results for the family of univariate distributions, allows to establish weak consistency and asymptotic normality of the estimators obtained through the method of “inference functions for margins”. The theoretical results are complemented by a simulation study and the practical use of the method is demonstrated on real data examples. A second part of the thesis consists of quantifying asymmetry and testing for it. In a univariate setting, there is a near unanimous agreement on the notion of skewness. Nevertheless, many more asymmetry measures exist, each with their benefits. Extending the concept of asymmetry to a multivariate setting is a much harder problem. Attempts have been made, but the unanimity of the univariate setting is no longer present. Most asymmetry measures are scalar or vector based measures, but this can lead to a loss of information concerning asymmetry. To this end, a novel functional asymmetry measure is proposed which is based on the natural idea of reflective symmetry around the mode. The proposed measure is also extended to the multivariate setting and a summarizing scalar (or vector based measure in a multivariate setting) is derived from it. Even though asymmetry can be measured, the question still poses itself whether the detected asymmetry is significant enough to describe the data as being asymmetric. Knowing if (a)symmetry is indeed present in data can help in making more accurate assumptions and in turn improve results. Be it e.g. through more accurate estimation of the center or choosing more suited models. Univariate symmetry is a widely known and deeply explored topic, but in a multivariate setting, the notion of symmetry is less clear as different views exist. In Chapter 5 the focus is on multivariate central symmetry. Besides existing tests for central symmetry, also a generalized likelihood ratio test which can be used for parametric, semi-parametric and non-parametric models is taken into account. In this, families of distributions which are asymmetric by nature but also contain their symmetric counterpart are used. A key focus is on skew-elliptical copulas, in particular the skew-normal copula. The concepts provided are demonstrated on real data and the performance of tests for central symmetry is empirically checked through an exploratory simulation study.