Page 430 - Contributed Paper Session (CPS) - Volume 4

CPS2523 Christian E. Galarza et al.

On moments of folded and truncated
multivariate extended skew-normal distributions
3
1
2
Christian E. Galarza ∗, Larissa Avila Matos , Victor Lachos Davila
1 Department of Statistics, Campinas State University, Campinas, Brazil.
2
Department of Statistics, Campinas State University, Campinas, Brazil.
3 Department of Statistics, University of Connecticut, Storrs, CT, USA.

Abstract
Following Kan & Robotti (2017), this paper develops recurrence relations for
integrals that involve the density of multivariate extended skew-normal
distributions, which includes the well-known skew-normal distribution
introduced by Azzalini & Dalla-Valle (1996) and the popular multivariate
normal distribution. These recursions offer fast computation of arbitrary order
product moments of truncated multivariate extended skew-normal and folded
multivariate extended skew-normal distributions with the product moments
of the multivariate truncated skew-normal, folded skew-normal, truncated
multivariate normal and folded normal distributions as a by product. Finally,
from the application point of view, these moments open the way to propose
analytical expressions on the E-step of the Expectation-Maximization (EM)
algorithm for complex data, such as, asymmetric longitudinal data with
censored and/or missing observations. These new methods are provided to
practitioners in the R MomTrunc package

Keywords
Product moments, Truncated distributions, Censored models.

1. Introduction
In many applications, researches often generate a large number of
datasets with values restricted to fixed intervals. For example, variables such
as pH, grades, viral load in HIV studies and humidity in environmental studies,
have upper and lower bounds due to detection limits, and the support of their
densities is restricted to some given intervals. Thus, the necessity of studying
the truncated distributions along with their properties arises naturally. In this
context, there has been a growing interest in evaluating the moments of
truncated distributions. Also, these variable are often skewed, departing from
the traditional assumption of using symmetric distributions. From Tallis (1961)
to Arismendi (2013), several works have pursued to compute formulae for the
rst two moments as well as higher order moments of truncated
univariate/multivariate distributions as the truncated normal (TN), truncated t-
Student (TT), truncated skew-normal (SN) (Azzalini & Dalla-Valle, 1996)
distributions among others. Main applications involve environmental studies,

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