Characteristics of multivariate distributions and the invariant coordinate system

Abstract

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.

On Invariant Coordinate System (ICS) Functionals

“…For the asymptotics, it is therefore not a restriction to assume that is a random sample from a distribution F with and where the diagonal elements of are λ 1 ≥⋯≥λ d > 0. Ilmonen et al (2010) then proved that if all the diagonal elements of are distinct and if then With a tiny modification in the proof of the above results in Ilmonen et al (2010), one can show that the three equations above are in fact true if λ i is distinct from all the other eigenvalues λ j , j ≠ i . The limiting joint distributions of the sample eigenvectors and sample eigenvalues for a subset with distinct population eigenvalues can then be derived from the limiting distributions of and .…”

“…If and are scatter functionals and if then the functional is an ICS functional in . See Ilmonen et al (2010) and Nordhausen et al (2010).…”

Blind Source Separation Based on Joint Diagonalization in R: The Packages JADE and BSSasymp

“…The statistical properties of FOBI are studied in Ilmonen, Nevalainen, and Oja (2010a) and Miettinen et al (2015b).…”

“…To compute (6) we first compute the asymptotic covariance matrices of the unmixing matrix estimatesŴ . Since all three independent components in the model have finite eighth moments, all three estimates have a limiting multivariate normal distribution (Ilmonen et al 2010a;Miettinen et al 2015b). The functions ASCOV_FOBI and ASCOV_JADE compute the asymptotic covariance matrices of the corresponding unmixing matrix estimatesŴ and the mixing matrix estimatesŴ −1 .…”

Vector-valued skewness for model-based clustering

“…A detailed description of ICS is beyond the scope of this paper, and the interested reader may refer to Tyler et al (2009), Nordhausen et al (2011 and Ilmonen et al (2010Ilmonen et al ( , 2012. One main advantage of ICS is its flexibility in discovering cluster structures by means of kurtosis, as measured by comparing two scatter matrices.…”