Estimation and Testing of Parameters in Multivariate Laplace Distribution

Abstract

In this article, we consider a multivariate Laplace distribution. When its skewness is zero, the distribution becomes a member of the elliptical family of distributions. We provide a test with its asymptotic null and nonnull distributions, for testing that the skewness is zero. Characteristics of the Laplace distribution such as mean, covariance matrix, third and fourth cumulants, and moments are given. Mardia's real-valued measures of skewness β_1p and kurtosis β_2p are defined in terms of cumulants, and an inequality between the skewness and kurtosis—namely, β_2p  ≥ p ² + β_1p , where p is the dimension of the random vector—is given. When p = 1, this reduces to the well-known inequality in the univariate case.

Key Words:

• Characteristic function
• Kurtosis
• Matrix derivative
• Multivariate cumulants
• Multivariate Laplace distribution
• Multivariate moments
• Skewness

Acknowledgments

The authors are thankful to Professor Samuel Kotz for drawing our attention to the multivariate Laplace distribution. We are also thankful to a referee for valuable comments and suggestions that have considerably improved the presentation and to Kaire Ruul for carrying out the simulation experiment. Tõnu Kollo is indebted to the Estonian Research Foundation (grant no. 4366 and object-oriented grant no. 0521s98) and to the Department of Statistics, University of Toronto for support.