Page 431 - Contributed Paper Session (CPS) - Volume 4

CPS2523 Christian E. Galarza et al.
macroeconomics and actuarial data. For instance, Arismendi (2013) provided
explicit expressions for computing arbitrary order product moments up to
order 4 of the TMN distribution by using the moment generating function
(MGF). However, the calculation of this approach relies on differentiation of
the MGF and can be somewhat time consuming.
Instead of differentiating the MGF of the TN distribution, Kan & Robotti
(2017) recently presented recurrence relations for integrals that involve
directly the density of the multivariate normal (MN) distribution for computing
arbitrary order product moments of the TN distribution. Although some
proposals to calculate the moments of the univariate truncated skew-normal
distribution (Flecher et al., 2010) and truncated univariate skew-
normal/independent distribution (Flecher et al., 2010) has recently been
published, so far, to the best of our knowledge, there is no attempt on
studying neither moments nor product moments of the folded multivariate
extended skew-normal (FESN) and truncated multivariate extended skew-
normal (TESN) distributions. Moreover, this approach allows to compute as a
by-product the moments of folded and truncated distributions, of the N (Kan
& Robotti, 2017), SN (Azzalini & Dalla-Valle, 1996), and its respective
univariate versions. The proposed algorithm and methods are implemented in
the new R MomTrunc package.
Over the last decade or so, censored modelling approaches have been
used in various ways to accommodate increasingly complicated applications.
Many of these extensions involve using N and its symmetrical extensions,
however statistical models based in distributions to accommodate censored,
missing and skewness, simultaneously, have remained relatively unexplored in
the statistical literature from the likelihood-based perspective. The results of
this paper allow, for instance, to derive analytical expressions on the E-step of
the EM algorithm for multivariate SN responses with censored and/or missing
observation.
The rest of this paper is organized as follows. In Section 2 we briefy discuss
some preliminary results related to the multivariate ESN, TESN and FESN
distributions and some of its key properties. Section 3 presents a recurrence
formula of an integral for the essential evaluation of moments of the TESN
distributions. Explicit expressions for the rst two moments of the TESN
distribution are also presented. In Section 4, we finally present interesting
results for the FESN distribution as well as explicit expressions for the
univariate case. Some concluding remarks are presented in Section 4. Proofs
and two interesting applications have been omitted due to the lack of space.

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