The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).
Key words and phrases
Moment generating function skewness stochastic representation quadratic form multivariate normal distribution Helmert matrix
This is a preview of subscription content, log in to check access.
Aigner, D. J., Lovell, C. H. K. and Schmidt, P. (1977). Formulation and estimation of stochastic frontier production function model,Journal of Econometrics,12, 21–37.CrossRefMathSciNetGoogle Scholar
Azzalini, A. (1985). A class of distribution which includes the normal ones,Scandinavian Journal of Statistics,12, 171–178.MathSciNetMATHGoogle Scholar
Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution.Journal of the Royal Statistical Society. Series B. Statistical Methodology,3, 579–602.MathSciNetGoogle Scholar
Genton, M. G., He, L. and Liu, X. (2001). Moments of skew-normal random vectors and their quadratic forms.Statistics & Probability Letters,51, 319–325.CrossRefMathSciNetMATHGoogle Scholar
Gupta, A. K. and Chen, T. (2001). Goodness of fit tests for the skew-normal distribution,Communication in Statistics, Computation and Simulation,30 (4), 907–930.CrossRefMathSciNetMATHGoogle Scholar
Gupta, A. K. and Chen, J. T. (2003): On the sample characterization criterion for normal distributions,Journal of Statistical Computation and Simulation,73, 155–163.CrossRefMathSciNetMATHGoogle Scholar