Tõnu Kollo

Advanced multivariate statistics with matrices

Basic Matrix Theory and Linear Algebra.- Multivariate Distributions.- Distribution Expansions.- Multivariate Linear Models.

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. Mardias real-valued measures of skewness β1p and kurtosis β2p are defined in terms of cumulants, and an inequality between the skewness and kurtosis—namely, β2p ≥ p 2 + β1p , 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.

Parameter Estimation and Application of the Multivariate Skew t -Copula

Copula theory has got a rapid development in recent years. Most used copulas are symmetric: Archimedean are symmetric by construction while other continuous multivariate copulas are usually constructed from elliptical distributions and therefore are symmetric. From skewed copulas we can refer only to a copula introduced in [5], which the authors called skew t-copula. The construction of it differs from our approach.We introduce a multivariate t-copula which is based on the skew t-distribution introduced in [1]. Parameters of the copula have been estimated by method of moments and a simulation rule is given. The behaviour of estimates of the shape parameter of the skewed t-distribution is illustrated by simulation. The skew t-copula is used for modelling real data.

Approximations to the distribution of the sample correlation matrix

In this article, multivariate density expansions for the sample correlation matrix R are derived. The density of R is expressed through multivariate normal and through Wishart distributions. Also, an asymptotic expansion of the characteristic function of R is derived and the main terms of the first three cumulants of R are obtained in matrix form. These results make it possible to obtain asymptotic density expansions of multivariate functions of R in a direct way.

Approximating by the Wishart distribution

Approximations of density functions are considered in the multivariate case. The results are presented with the help of matrix derivatives, powers of Kronecker products and Taylor expansions of functions with matrix argument. In particular, an approximation by the Wishart distribution is discussed. It is shown that in many situations the distributions should be centred. The results are applied to the approximation of the distribution of the sample covariance matrix and to the distribution of the non-central Wishart distribution.

The derivative of an orthogonal matrix of eigenvectors of a symmetric matrix

Abstract The authors supply the derivative of an orthogonal matrix of eigenvectors of a real symmetric matrix. To illustrate the applicability of their result they consider a real symmetric random matrix for which a more or less standard convergence in distribution is assumed to hold. The well-known delta method is then used to get the asymptotic distribution of the orthogonal eigenmatrix of the random matrix.

Approximation of the non-null distribution of generalized T2-statistics

Abstract The non-null distributions of Hotellings T 2 -statistic and a generalized F-statistic [Biometrics 52 (1996) 964] are approximated by asymptotic normal distributions. The distributions are derived under assumption of ellipticity of the population from where results for the normal population also follow. The main term of the bias of Hotellings T 2 -statistic is found in the case of normal population. A simulation experiment is carried out and its results presented. In Appendix A, derivatives of a rectangular matrix of eigenvectors of a symmetric matrix and corresponding eigevalue matrix are found.

From Multivariate Skewed Distributions to Copulas

In this paper, a methodology is presented for constructing skewed multivariate copulas to model data with possibly different marginal distributions. Multivariate skew elliptical distributions are transformed into corresponding copulas in the similar way as the Gaussian copula and the multivariate t-copula are constructed. Three-parameter skew elliptical distributions are under consideration. For parameter estimation of the skewed distributions, the method of moments is used. To transform mixed third-order moments into a parameter vector, the star product of matrices is used; for star product and its applications, see, for example, Kollo (J. Multivar. Anal. 99:2328–2338, 2008) or Visk (Commun. Stat. 38:461–470, 2009). Results of the first applications are shortly described and referred to.

Risk Modeling for Future Cash Flow Using Skew t-Copula

In this article, we estimate future annual cash flow of an insurance company. Data of three main different business lines of the company are used to predict the next years cash flow. The individual business lines are modelled by Gamma, Pareto and lognormal distributions. Daily payments were correlated and therefore the joint distribution of the business lines was found. As the model, the skew t-copula introduced in Kollo and Pettere (2010) was used. Simulation from the joint distribution was carried out to predict the next years cash flow. The obtained results were compared with the case when the dependencies between the business lines were not taken into account. Presented model gives stochastic approach for future payments.

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.