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B1620
Title: Testing dimensionality of multivariate variance components Authors:  Satoshi Kuriki - The Institute of Statistical Mathematics (Japan) [presenting]
Tomoyuki Shirai - Kyushu University (Japan)
Khanh Duy Trinh - Kyushu University (Japan)
Abstract: Let A be a standard Gaussian random matrix in the space Sym(n) of n by n real-symmetric or Hermitian matrices (i.e., GOE or GUE, respectively). Let PD(n) be the cone of positive semidefinite matrices in Sym(n). We derive the distribution of the squared distance between the random matrix A and the cone PD(n). This distribution appears in balanced multivariate variance components model. In this model, within and between sum of squares matrices (H and G, say) are independent Wishart matrices. The difference of matrix parameters of H and G is a positive semidefinite matrix referred to as variance components. The rank of the variance components matrix is typically much smaller than the size of the matrices H and G, and its inference is crucial in modeling. When the number of groups goes to infinity, the degrees of freedom of H and G go to infinity, and then the LRT for the rank of the variance components matrix converges to the squared distance between A and PD(n). In GOE and GUE cases, the distributions are shown to be mixtures of chi-square distributions with weights expressed in terms of the Pfaffian or the determinant. Moreover, when the matrix size n goes to infinity, the limiting distribution is proved to be Gaussian. This Gaussianity was conjectured in previous literature. Based on the obtained distributions, we propose a multiple testing procedure to estimate the rank of the variance components matrix. Mouse growth data are analyzed as an example.