SUMMARY The paper extends earlier work on the so-called skew-normal distribution, a family of distributions including the normal, but with an extra parameter to regulate skewness. The present work introduces a multivariate parametric family such that the marginal densities are scalar skew-normal, and studies its properties, with special emphasis on the bivariate case.

/pdf/the-multivariate-skew-normal-distribution-265gie4y64.pdf

The multivariate skew-normal distribution

Azzalini and Dalla Valle have recently discussed the multivariate skew normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.

Statistical applications of the multivariate skew normal distribution

In this article, we present a concise review of developments on various continuous multivariate distributions. We first present some basic definitions and notations. Then, we present several important continuous multivariate distributions and list their significant properties and characteristics.


Keywords:

generating function;
moments;
conditional distribution;
truncated distribution;
regression;
bivariate normal;
multivariate normal;
multivariate exponential;
multivariate gamma;
dirichlet;
inverted dirichlet;
liouville;
multivariate logistic;
multivariate pareto;
multivariate extreme value;
multivariate t;
wishart translated systems;
multivariate exponential families

/pdf/continuous-multivariate-distributions-1gnnj8yohf.pdf

Continuous Multivariate Distributions

Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.

Statistical applications of the multivariate skew-normal distribution

This paper considers the estimation of Kumbhakar et al. (J Prod Anal. doi:
 10.1007/s11123-012-0303-1
 , 2012) (KLH) four random components stochastic frontier (SF) model using MLE techniques. We derive the log-likelihood function of the model using results from the closed-skew normal distribution. Our Monte Carlo analysis shows that MLE is more efficient and less biased than the multi-step KLH estimator. Moreover, we obtain closed-form expressions for the posterior expected values of the random effects, used to estimate short-run and long-run (in)efficiency as well as random-firm effects. The model is general enough to nest most of the currently used panel SF models; hence, its appropriateness can be tested. This is exemplified by analyzing empirical results from three different applications.

Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency

The single variable marginal density function of a truncated multivariate normal density function is derived in a form that can be evaluated using an available computer algorithm. It is shown that the marginal density function is a truncated normal density function multiplied by a “skew function”

Jack Cartinhour

Papers

One-dimensional marginal density functions of a truncated multivariate normal density function

An iterative time-frequency domain algorithm for reduction of peak-to-rms ratio