On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation

Abstract

Multivariate normal mean–variance mixture (NMVM) distributions are alternatives to the multivariate normal distribution when, in practice, we encounter data sets possessing large skewness and/or kurtosis measures. In this paper, we focus on truncated forms of NMVM distributions and derive explicit expressions for the first two moments. Our results are general which can be applied for any NMVM distribution. In particular, we derive explicit expressions for the first two moments of doubly truncated multivariate generalized hyperbolic (GH) distribution. We show that by using the results established here, the multivariate tail conditional expectation (MTCE) can be obtained for any NMVM distribution.