. 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 Skew-normal Distribution and Related Multivariate Families.

Preface 1. Modulation of symmetric densities 2. The skew-normal distribution: probability 3. The skew-normal distribution: statistics 4. Heavy and adaptive tails 5. The multivariate skew-normal distribution 6. Skew-elliptical distributions 7. Further extensions and other directions 8. Application-oriented work Appendices References.

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The Skew-Normal and Related Families

The univariate skew-normal distribution was introduced by Azzalini in 1985 as a natural extension of the classical normal density to accommodate asymmetry. He extensively studied the properties of this distribution and in conjunction with coauthors, extended this class to include the multivariate analog of the skew-normal. Arnold et al. (1993) introduced a more general skew-normal distribution as the marginal distribution of a truncated bivariate normal distribution in whichX was retained only ifY satisfied certain constraints. Using this approach more general univariate and multivariate skewed distributions have been developed. A survey of such models is provided together with discussion of related inference questions.

Skewed multivariate models related to hidden truncation and/or selective reporting

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Improvement by Means of Selection

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main prop erties of selection distributions and show how various families of multivariate skewed distributions, such as the skew-normal and skew-elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms. Une perspective integree des lois asymetriques issues de processus de selection Resume: Les familles param6triques de lois multivari6es non gaussiennes ont suscite beaucoup d'int6ret depuis quelques decennies. Les auteurs proposent une nouvelle definition du concept de loi de selection qui englobe plusieurs familles connues de lois asymetriques multivari6es. Leurs travaux sont motives par diverses situations faisant intervenir des mecanismes de selection et conduisant 'a des lois asymetriques. Ils mentionnent les principales propri6t6s des lois de selection et montrent comment diverses familles de lois asymetriques multivariees telles que les lois asymetriques normales ou elliptiques emergent comme cas particuliers. Les auteurs presentent en outre plusieurs methodes de construction de lois de selection fondees sur des mecanismes lin6aires ou non lineaires.

A unified view on skewed distributions arising from selections

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Z. W. Birnbaum

Papers

Effect of Linear Truncation on a Multinormal Population