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Unformatted text preview: A Constructive Representation of Univariate Skewed Distributions Jos´ e T.A.S. Ferreira and Mark F.J. Steel * Department of Statistics University of Warwick, UK Abstract We introduce a general perspective on the introduction of skewness into symmetric distributions. Making use of inverse probability integral transformations we provide a constructive representation of skewed distributions, where the skewing mechanism and the original symmetric distributions are specified separately. We study the effects of the skewing mechanism on e.g. modality, tail behaviour and the amount of skewness generated. In light of the constructive representation, we review a number of characteristics of three classes of skew distributions previously defined in the literature. The representation is also used to introduce two novel classes of skewed distributions. Finally, we incorporate the different classes of distributions into a Bayesian linear regression framework and analyse their differences and similarities. Keywords : Arnold and Groeneveld skewness measure, Bayesian regression model, inverse proba bility integral transformation, modality, skewing mechanism, tail behaviour 1 Introduction Recent years have seen a resurgent interest in the theory and application of distributions that can account for skewness. This article studies skewness in univariate data and its objective is threefold. First, we present a general constructive representation of univariate skewed distributions. We then use this representation to study classes of such distributions previously proposed in the literature and to construct novel classes. Finally, we use these classes in Bayesian regression modelling. The most common approach to the creation of skewed distributions, and the one we are interested in here, is to introduce skewness into an originally symmetric distribution. This approach underlies the general classes of skewed distributions generated, for example, by hidden truncation models (see e.g. Azzalini, 1985 and Arnold and Beaver, 2002), inverse scale factors in the positive and the negative orthant (Fern´andez and Steel, 1998) and, more recently, order statistics (Jones, 2004). The key advantage of skewing a symmetric distribution F is that in doing so it is possible to retain some of the properties of F , which are often well known. All methods mentioned in the previous paragraph keep a subset of these properties, with distinct models leading to distinct subsets. Here we propose a unified perspective on skewed distributions. The idea is to separate the skewing mechanism from the symmetric distribution that serves as a starting point. This is appealing both methodologically and in the context of applications. By separating the two components, different * Address for correspondence : Mark Steel, Department of Statistics, University of Warwick, Coventry, CV4 7AL, U.K....
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 Fall '00
 Akritas
 Statistics, Normal Distribution, skewed distributions, skewing mechanism

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