Antonella Capitanio

Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution

Summary. A fairly general procedure is studied to perturb a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is sufficiently general to encompass some recent proposals in the literature, variously related to the skew normal distribution. The special case of skew elliptical densities is examined in detail, establishing connections with existing similar work. The final part of the paper specializes further to a form of multivariate skew t-density. Likelihood inference for this distribution is examined, and it is illustrated with numerical examples.

Statistical applications of the multivariate skew normal distribution

Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.

Graphical models for skew-normal variates

This paper explores the usefulness of the multivariate skew-normal distribution in the context of graphical models. A slight extension of the family recently discussed by Azzalini & Dalla Valle (1996) and Azzalini & Capitanio (1999) is described, the main motivation being the additional property of closure under conditioning. After considerations of the main probabilistic features, the focus of the paper is on the construction of conditional independence graphs for skew-normal variables. Necessary and sufficient conditions for conditional independence are stated, and the admissible structures of a graph under restriction on univariate marginal distribution are studied. Finally, parameter estimation is considered. It is shown how the factorization of the likelihood function according to a graph can be rearranged in order to obtain a parameter based factorization.

On the approximation of the tail probability of the scalar skew-normal distribution

SummaryIn this note the tail behaviour of the univariate skew-normal distribution is studied. It is shown, in particular, that the rate of decay to zero of the right and left tail probability are different, one being equal to the normal one and the other thinner. Asymptotic approximations for right and left tail probability are also given.

Modulation of symmetric densities

Motivation This book deals with a formulation for the construction of continuous probability distributions and connected statistical aspects. Before we begin, a natural question arises: with so many families of probability distributions currently available, do we need any more? There are three motivations for the development ahead. The first motivation lies in the essence of the mechanism itself, which starts with a continuous symmetric density function that is then modified to generate a variety of alternative forms. The set of densities so constructed includes the original symmetric one as an ‘interior point’. Let us focus for a moment on the normal family, obviously a case of prominent importance. It is well known that the normal distribution is the limiting form of many non-normal parametric families, while in the construction to follow the normal distribution is the ‘central’ form of a set of alternatives; in the univariate case, these alternatives may slant equally towards the negative and the positive side. This situation is more in line with the common perception of the normal distribution as ‘central’ with respect to others, which represent ‘departures from normality’ rather than ‘incomplete convergence to normality’. The second motivation derives from the applicability of the mechanism to the multivariate context, where the range of tractable distributions is much reduced compared to the univariate case. Specifically, multivariate statistics for data in Euclidean space is still largely based on the normal distribution.

The Skew-Normal and Related Families: Heavy and adaptive tails

Motivating remarks The skew-normal density has very short tails. In fact, the rate of decay to 0 of the density φ( x ; α) as | x | → ∞ is either the same as the normal density or even faster, depending on whether x and α have equal or opposite sign, as specified by Proposition 2.8. This behaviour makes the skew-normal family unsuitable for a range of application areas where the distribution of the observed data is known to have heavier tails than the normal ones, sometimes appreciably heavier. To construct a family of distributions of type (1.2) whose tails can be thicker than a normal ones, a solution cannot be sought by replacing the term Φ;(α x ) in (2.1) with some other term G 0 { w(x )}, since essentially the same behaviour of the SN tails would be reproduced. The only real alternative is to adopt a base density f 0 in (1.2) with heavier tails than the normal density. For instance, we could select the Laplace density exp(−| x |)/2, whose tails decrease at exponential rate, to play the role of base density and proceed along lines similar to the skew-normal case. This is a legitimate program, but it is preferable that f 0 itself is a member of a family of symmetric density functions, depending on a tail weight parameter, v say, which allows us to regulate tail thickness. For instance, one such choice for f 0 is the Students t family, where v is represented by the degrees of freedom.