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Register a new userAsymptotic power of Raos score test for independence in high dimensions
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Dennis Leung
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Let ${bf R}$ be the Pearson correlation matrix of $m$ normal random variables. The Raos score test for the independence hypothesis $H_0 : {bf R} = {bf I}_m$, where ${bf I}_m$ is the identity matrix of dimension $m$, was first considered by Schott (2005) in the high dimensional setting. In this paper, we study the asymptotic minimax power function of this test, under an asymptotic regime in which both $m$ and the sample size $n$ tend to infinity with the ratio $m/n$ upper bounded by a constant. In particular, our result implies that the Raos score test is rate-optimal for detecting the dependency signal $|{bf R} - {bf I}_m|_F$ of order $sqrt{m/n}$, where $|cdot|_F$ is the matrix Frobenius norm.
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We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both m and n tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendalls tau and a dominating term of Spearmans rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffdings $D$, or the $tau^*$ of Bergsma and Dassios (2014). As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under Gaussian equicorrelation alternatives as well as in numerical experiments for specific cases of more general alternatives.
This paper proposes a new statistic to test independence between two high dimensional random vectors ${mathbf{X}}:p_1times1$ and ${mathbf{Y}}:p_2times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of ${mathbf{X}}$ and ${mathbf{Y}}$. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when $p_1$ and $p_2$ are both comparable to the sample size $n$. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.
Deheuvels [J. Multivariate Anal. 11 (1981) 102--113] and Genest and R{e}millard [Test 13 (2004) 335--369] have shown that powerful rank tests of multivariate independence can be based on combinations of asymptotically independent Cram{e}r--von Mises statistics derived from a M{o}bius decomposition of the empirical copula process. A result on the large-sample behavior of this process under contiguous sequences of alternatives is used here to give a representation of the limiting distribution of such test statistics and to compute their relative local asymptotic efficiency. Local power curves and asymptotic relative efficiencies are compared under familiar classes of copula alternatives.
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these results, we develop a new statistic to test the equality of two spiked population covariance matrices. Numerical studies show that the new test procedure is more powerful than some existing methods.
We introduce a new test procedure of independence in the framework of parametric copulas with unknown marginals. The method is based essentially on the dual representation of $chi^2$-divergence on signed finite measures. The asymptotic properties of the proposed estimate and the test statistic are studied under the null and alternative hypotheses, with simple and standard limit distributions both when the parameter is an interior point or not.
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