2003, Volume 65, Pt. 4 , 821--835

**DENSITY EXPANSIONS BASED ON THE
MULTIVARIATE SKEW NORMAL DISTRIBUTION**

By

ARJUN K. GUPTA, Bowling Green State University, USA and T\~ONU KOLLO, University of Tartu, Estonia

SUMMARY. In this paper the multivariate skew normal distribution, introduced by Azzalini and Dalla Valle (1996), is used as a basis in density expansions. A short summary of main properties of the distribution is given and the first three derivatives of the multivariate skew normal density function are presented. These results are used in a general formal density expansion through the multivariate skew normal distribution. The general density expansion is specified for some choices of parameters, the results are illustrated on an example where the density function of the largest eigenvalue of the sample dispersion matrix is approximated.

*AMS (1991) subject classification. *Primary 62H10; secondary 62E17.

*Key words and phrases. *Skew normal distribution, moment generating
function, multivariate cumulants and moments, multivariate density expansion.