Abstract. A new class of density functions depending on a shape parameter is introduced, such that the value 0 for this parameter corresponds to the standard normal density. The properties of this class of density functions are studied.
Abstract.
Further results are presented about a class of density functions
considered by the author in a previous paper (1985). In particular,
an additional shape parameter is introduced which allows a wide
range for the indices of skewness and curtosis.
Note: an
errata-corrige is available.
Abstract. The paper extends earlier work on the so-called skew-normal distribution, a family of distributions including the normal, but with an extra parameter to regulate skewness. The present work introduces a multivariate parametric family such that the marginal densities are scalar skew-normal, and studies its properties, with special emphasis on the bivariate case.
Abstract. Azzalini & Dalla Valle (1996) 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.
Abstract. The problem of finding the smallest region with given probability mass is considered for the case of a multivariate random variable with skew-normal distribution. A simple but accurate approximate solution is proposed.
Abstract. A fairly general procedure is studied to perturbate a multivariate density satisfying a weak form of multivariate symmetry, and to generate a whole set of non-symmetric densities. The approach is general enough to encompass a number of 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.
Abstract. 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.
Abstract. The U.S. family income data for the years 1970, 1975, 1978, 1980, 1985 and 1990 was fitted using the log-normal, Gamma, Singh--Maddala, Dagum type I and generalized Beta of second kind distributions, among others, in earlier publications. Here we supplement these fittings by adding the log-skew-normal and log-skew-t distributions. In addition, we have performed similar numerical comparisons using 1997 income data collected in a sample survey from several European countries. The overall picture emerging from these numerical comparisons indicates that, while the log-skew-normal distribution provides a somewhat variable degree of goodness-of-fit, the log-skew-t distribution seems to fit the data satisfactorily in a quite consistent way, and on the par with the most creditable distributions.
Abstract. The stress-strength model is considered in the case of skew-normal variates. Some exact probability results are given. For the case that either the stress or the strength has a skew-normal distribution, inferential issues are considered in a likelihood context, and simulation results provided which indicate a satisfactory agreement of nominal and actual confidence level for interval estimation.
Abstract. This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so-called skew-normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew-elliptical families, and some forms of semi-parametric formulations, extending the relevance of the field much beyond the original theme of `skewness'. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non-linear time series, especially in connection with financial data, and more.
The paper is followed by a discussion by Marc Genton (pp. 189-198) and the author (pp. 199-200).
Abstract. The distribution theory literature connected to the multivariate skew-normal distribution has grown rapidly in recent years, and a number of extensions and alternative formulations have been put forward. Presently there are various coexisting proposals, similar but not identical, and with rather unclear connections. The purpose of this paper is to unify these proposals under a new general formulation, clarifying at the same time their relationships. The final part sketches an extension of the argument to the skew-elliptical family.
Abstract. In neuropsychological single-case research inferences concerning a patientâs cognitive status are often based on referring the patient's test score to those obtained from a modestly sized control sample. Two methods of testing for a deficit (z and a method proposed by Crawford and Howell [Crawford, J. R. & Howell, D. C. (1998). Comparing an individualâs test score against norms derived from small samples. The Clinical Neuropsychologist, 12, 482-486]) both assume the control distribution is normal but this assumption will often be violated in practice. Monte Carlo simulation was employed to study the effects of leptokurtosis and the combination of skew and leptokurtosis on the Type I error rates for these two methods. For Crawford and Howell's method, leptokurtosis produced only a modest inflation of the Type I error rate when the control sample N was small-to-modest in size and error rates were lower than the specified rates at larger N. In contrast, the combination of leptokurtosis and skew produced marked inflation of error rates for small Ns. With a specified error rate of 5%, actual error rates as high as 14.31% and 9.96% were observed for z and Crawford and Howellâs method respectively. Potential solutions to the problem of non-normal data are evaluated.
Abstract. For statistical inference connected to the scalar skew-normal distribution, it is known that the so-called centred parametrization provides a more convenient parametrization than the one commonly employed for writing the density function. We extend the definition of the centred parametrization to the multivariate case, and study the corresponding information matrix.
Abstract. The robustness problem is tackled by adopting a parametric class of distributions flexible enough to match the behaviour of the observed data. In a variety of practical cases, one reasonable option is to consider distributions which include parameters to regulate their skewness and kurtosis. As a specific representative of this approach, the skew-$t$ distribution is explored in more detail, and reasons are given to adopt this option as a sensible general-purpose compromise between robustness and simplicity, both of treatment and of interpretation of the outcome. Some theoretical arguments, outcomes of a few simulation experiments and various wide-ranging examples with real data are provided in support of the claim.
Abstract. An active stream of literature has followed up the idea of skew-elliptical densities initiated by Azzalini and Capitanio (1999). Their original formulation was based on a general lemma which is however of broader applicability than usually perceived. This note examines new directions of its use, and illustrates them with the construction of some probability distributions falling outside the family of the so-called skew-symmetric densities.
Abstract. In the context of clinical trials where one of several doses or treatments is selected in a phase II study to be examined further in a phase III study, we develop a formulation for the combination of the overall information obtained from such studies, which mimics the logic followed in actual drug development. The associated distribution theory is exact under the normality assumption. Extensions to more complex situations are sketched briefly.
Abstract. We develop estimating equations for the parameters of the base density of a skewsymmetric distribution. The method is based on an invariance property with respect to asymmetry. Various properties of this approach and the selection of a root are discussed. We also present several extensions of the methodology, namely to the regression setting, the multivariate case, and the skew-t distribution. The approach is illustrated on several simulations and a numerical example.
Abstract. The skew-t family, in its univariate and multivariate versions, is a parametric family of probability distributions which is currently under intense investigation because of several appealing properties. The present paper addresses the question of the choice of its parameterization, and more generally of the selection of quantities of interest associated to this distribution.
Abstract. The family of skew-symmetric distributions is a wide set of probability density functions obtained by suitably combining a few components which can be quite freely selected provided some simple requirements are satisfied. Although intense recent work has produced several results for certain sub-families of this construction, much less is known in general terms. The present paper explores some questions within this framework, and provides conditions for the above-mentioned components to, ensure that the final distribution enjoys specific properties.