Abstract
This paper is concerned with the multivariate extended skew-normal [MESN] and multivariate extended skew-Student [MEST] distributions, that is, distributions in which the location parameters of the underlying truncated distributions are not zero. The extra parameter leads to greater variability in the moments and critical values, thus providing greater flexibility for empirical work. It is reported in this paper that various theoretical properties of the extended distributions, notably the limiting forms as the magnitude of the extension parameter, denoted τ in this paper, increases without limit. In particular, it is shown that as τ→−∞, the limiting forms of the MESN and MEST distributions are different. The effect of the difference is exemplified by a study of stockmarket crashes. A second example is a short study of the extent to which the extended skew-normal distribution can be approximated by the skew-Student.
Highlights
The skew-normal distribution was introduced in [1] and the skew-Student in [2].These two distributions share the property that they may be derived formally
The main aim of this paper is to present properties of the multivariate extended skew-normal (MESN) and multivariate extended skew-Student (MEST) distributions
In their simple form, hidden truncation models are concerned with the bivariate normal distribution of ( X, Y ) in situations in which X is observed if Y is greater than a given threshold, here denoted τ
Summary
The skew-normal distribution was introduced in [1] and the skew-Student in [2]. These two distributions share the property that they may be derived formally. Use of extended versions of the skew-normal or skew-Student gives greater variability in the moments and critical values of the distributions. For empirical applications, this offers the possibility of better model fit. For a specified univariate application, it would be straightforward to estimate the parameters of both distributions and make an informed choice using a test of fit or, for example, consideration of the tails of the distribution Such an alternative may be attractive, but the suggestion could well be made in reverse: the extended skew-normal could be an alternative to the skew-Student. Notation not defined explicitly in the text is that in common use
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