Characteristics of multivariate distributions and the invariant coordinate system

Pauliina Ilmonen (1) , Jaakko Nevalainen (2) , Hannu Oja (1)

1 University of Tampere [Finland]
2 Statistics/Department of Social Research

We consider a semiparametric multivariate location-scatter model where the standardized random vector of the model is fixed using simultaneously two location vectors and two scatter matrices. The approach using location and scatter functionals based on the first four moments serves as our main example. The four functionals yield in a natural way the corresponding skewness, kurtosis and unmixing matrix functionals. Affine transformation based on the unmixing matrix transforms the variable to an invariant coordinate system. The limiting properties of the skewness, kurtosis, and unmixing matrix estimates are derived under general conditions. We discuss related statistical inference problems, the role of the sample statistics in testing for normality and ellipticity, and connections to invariant coordinate selection and independent component analysis.

Mots clés: 62H10; 62H12; 62G05; 62G20; 62F12; Asymptotic normality; Independent component analysis; Invariant coordinate selection; Multivariate kurtosis; Multivariate skewness; Scatter matrix

Domaines: Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Pauliina Ilmonen, Jaakko Nevalainen, Hannu Oja. Characteristics of multivariate distributions and the invariant coordinate system. Statistics and Probability Letters, 2010, 80 (23-24), pp.1844. ⟨10.1016/j.spl.2010.08.010⟩. ⟨hal-00691780⟩