Doubly truncated expectation and variance of univariate generalized skew-elliptical distributions with applications

Abstract

In this article, doubly truncated expectation (DTE) and variance (DTV) of univariate generalized skew-elliptical (GSE) distributions are investigated. In addition, we present an alternative form of DTE and DTV for this class of distributions in terms of the hazard function. This class of distributions includes many skewing distributions, for instance, generalized skew-normal, skew-Student-t, skew-logistic, skew-Laplace, and skew-Pearson type VII distributions. Also, we define truncated generalized skew-elliptical distributions and give the relations between moments of truncated distributions and truncated moments of distributions. Specially, we use the EM algorithm to give maximum likelihood estimation of parameters for generalized skew-elliptical distributions. Further, we apply our results to present tail conditional expectation (TCE) and tail variance (TV) for GSE distributions. We also structure an optimal portfolio selection involving DTE and DTV, and give its optimal solution. As an illustrative example, DTE and DTV of a skew-normal random variable are estimated by the Monte Carlo method. Finally, we use real data to fit and select the best distributions, and analysis TCE and TV for the logarithm of adjusted price of three companies (stocks) from S & P (Standard & Poor’s) sectors.

Keywords:

• Doubly truncated expectation
• Doubly truncated variance
• EM algorithm
• Generalized skew-elliptical distribution
• Monte Carlo method
• Tail conditional expectation
• Tail variance

Acknowledgments

The authors thank the anonymous referees and the Editor for their helpful comments and suggestions, which have led to the improvement of this article.

Disclosure statement

The authors declare that they have no conflicts of interest.

Additional information

Funding

The research was supported by the National Natural Science Foundation of China (No. 12071251).