Truncated and Limited Skew-Normal and Skew-t Distributions: Properties and an Illustration

The family of skew-normal distributions was discussed formally and popularized by Azzalini (1985, 1986) and since then it has been discussed extensively in the literature. In this article, we consider the truncated and limited forms of skew-normal and skew-t distributions and discuss some of their mathematical properties. We also derive explicit expressions for their cumulative distribution functions, moment generating function, and moments. We then pay special attention to the truncated skew-Cauchy distribution and present the corresponding results. Finally, we illustrate the use of this model in fitting a real-data on the exchange rate between the UK pound and the US dollar.