On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

Abstract

By studying the family of p-dimensional scale mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Marčenko-Pastur law. A different and new limit is found and characterized. The reasons of failure of the Marčenko-Pastur limit in this situation are found to be a strong dependence between the p-coordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analize the traditional John's type test we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. It is shown that the John's test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used ICL and BIC criteria in detecting non spherical component covariance matrices of a high-dimensional mixture.

On singular values of data matrices with general independent columns

“…Finite mixture model. Models based on finite mixture distributions provides a flexible extension of classical statistical models and have been applied in diverse areas such as genetics, signal processing and machine learning [7,13,17]. The observations in a finite mixture model can always be viewed as samples drawn randomly from several populations with different means or covariance matrices with certain proportion.…”

“…The observations in a finite mixture model can always be viewed as samples drawn randomly from several populations with different means or covariance matrices with certain proportion. As a special case, a scale mixture model was studied by [13], in which covariance matrices of different populations differ only by a random factor. The dependence among LSDs of the sample covariance matrix and the common population covariance matrix as well as the distribution of scale variable is derived.…”

High-dimensional covariance matrices in elliptical distributions with application to spherical test

“…Indeed, Bai and Zhou (2008) proved that the SD F Bn of B n converges to a common generalized Marčenko-Pastur law almost surely if, for any sequence of symmetric matrices {C p } with bounded spectral norm, (1.5) V ar(x ′ C p x) = o(p 2 ) as p, n → ∞. This condition is also sharp for the convergence, see Li and Yao (2017) for an example. What is more, this condition holds for a list of well known elliptical distributions, such as multivariate normal distributions, multivariate Pearson type II distributions, power exponential distributions, and a more general family of multivariate Kotz-type distributions (Kotz, 1975), see Section 2 for more details.…”

“…We should note that the condition (2.1) excludes some elliptical distributions, such as multivariate student-t distributions and normal scale mixtures, as shown in the 5-6th rows of Table 1. Indeed, sample eigenvalues from these distributions do not obey the generalized Marčenko-Pastur law (El Karoui, 2009;Li and Yao, 2017), which are then out of the scope of this paper. Table 1 Some elliptical distributions and their verification of the condition (2.1).…”

Geometric consistency of principal component scores for high‐dimensional mixture models and its application

“…Borysov, Hannig, and Marron (2014) considered hierarchical clustering for high-dimensional data. Li and Yao (2018) considered a model-based clustering for a high-dimensional mixture. Given this background, we decided to focus on high-dimensional structures of multiclass mixture models via PCA.…”