Meelis Käärik

On Parametrization of Multivariate Skew-Normal Distribution

Skew-normal distribution is an extension of the normal distribution where the symmetry of the normal distribution is distorted with an extra parameter called the shape parameter. The scalar version of the skew-normal distribution was introduced already in works by Azzalini (1985) and by Henze (1986). The multivariate skew-normal distribution has received considerable attention over the last years, but unfortunately there is no unique straightforward generalization from the scalar case. Therefore, various families of skew-symmetric distributions with different properties have been proposed and studied. In our work we refer to the “classical” multivariate skew-normal distribution introduced by Azzalini and Dalla Valle (1996). It must be noted that even the Azzalini’s skew-normal distribution can be parametrized in many different ways starting from the initial -parametrization to the currently prevalent -parametrization. This motivated us to search for viable alternatives and compare the overall behavior of different parametrizations, the key problems addressed in our article.

Approximation of Distributions by Parametric Sets

Let P be a probability distribution on a separable metric space (S,d). We study the following problem of approximation of a distribution P by a set from a given class A⊂2S: W(A,P)≡∫Sϕ(d(x,A))P(dx)→min A∈A, where ϕ is a nondecreasing function. A special case where A is a parametric class A={A(Θ):Θ∈T} is considered in detail. Our main interest is to obtain convergence results for sequences {A*n}, where A*n is an optimal set for a measure Pn satisfying Pn⇒P, as n→∞.

Imputation by Gaussian Copula Model with an Application to Incomplete Customer Satisfaction Data

We propose the idea of imputing missing value based on conditional distributions, which requires the knowledge of the joint distribution of all the data. The Gaussian copula is used to find a joint distribution and to implement the conditional distribution approach.

Asymptotic normality of estimators for parameters of a multivariate skew-normal distribution

ABSTRACT In this paper, asymptotic normality is established for the parameters of the multivariate skew-normal distribution under two parametrizations. Also, an analytic expression and an asymptotic normal law are derived for the skewness vector of the skew-normal distribution. The estimates are derived using the method of moments. Convergence to the asymptotic distributions is examined both computationally and in a simulation experiment.

On the correlation structures of multivariate skew-normal distribution

Skew-normal distribution is an extension of the normal distribution where the symmetry of the normal distribution is distorted with an extra parameter. A multivariate skew-normal distribution has been parametrized differently to stress different aspects and constructions behind the distribution. There are several possible parametrizations available to define the skew-normal distribution. The current most common parametrization is through Ω and α , as an alternative, parametrization through Ω and δ can be used if straightforward relation to marginal distributions is of interest. The main problem with { Ω , δ }-parametrization is that the vector δ cannot be chosen independently of Ω . This motivated us to investigate what are the possibilities of choosing δ under different correlation structures of Ω . We also show how the assumptions on structure of δ and Ω affect the asymmetry parameter α and correlation matrix R of corresponding skew-normal random variable.