Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions

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 tau in this paper, increases without limit. In particular, it is shown that as tau -> -infinity, the limiting forms of the MESN and MEST distributions are different. The effect of the difference is exemplified by a study of stock market 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.