Some Tests Concerning the Covariance Matrix in High Dimensional Data

By Muni S. Srivastava. 2005

265 citations recorded by the source corpus.

Recent advances in analyzing high dimensional data with fewer observations require that certain assumptions made implicitly or explicitly in analyzing them should be ascertained. For example, Dudoit et al. (2002) in their analysis of microarrays data on genes assume that the covariance matrices are diagonal matrices, and thus their distance function uses only the diagonal elements of the sample covariance matrix. The good performance of their procedures appear to suggest that this indeed might be the case. To ascertain these assumptions on the covariance matrix, the likelihood ratio tests cannot be used as the sample size N = n+1 could be smaller than the dimension p. Although, the locally best invariant (LBI) test proposed by John (1971) and considered by Suguira (1972), and Nagao (1973) for the spherecity hypothesis and the LBI test given by Nagao (1973) for testing the hypothesis that the covariance matrix Σ is an identity matrix can be computed for all sample sizes, there appears to be no theoretical justification for using them as LBI tests cannot be obtained when n<p . Thus, we consider a distance function between the null hypothesis and the alternative hypothesis, and propose tests based on consistent estimators of these parametric functions of the covariance matrix Σ for testing the hypothesis of spherecity of the covarinace matrix, and for testing the hypothesis that the covarinace matrix is an identity matrix. In addition, under the same set of conditions, we provide tests for testing the hypothesis that the covarinace matrix is a diagonal matrix. Asymptotic distributions of these test statistics are given under the hypothesis as well as under the alternative hypothesis. Our focus is however for the case when n = O(p δ ), 0 <δ ≤ 1. Thus, it includes the case when (n/p) → 0. The

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9bee54581bb34d8639da62ebc82851cb08a13def

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Citations in the corpus

Optimal-order bounds on the rate of convergence to normality in the multivariate delta method (2009, result): “For instance, our results can easily yield BE bounds for statistics proposed by Srivastava [76] or Fisher et al. [18]; Chen et al. [12] propose a statistic for the sphericity test which is a function of a U -statistic, for which the methods of this paper and [11] could presumably be adapted.”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “Theorem 11 (Sun and Zhang (2012)) Suppose that X is Gaussian and cn,p ( log p n )1/2 = o(1).”
Multivariate-Sign-Based High-Dimensional Tests for the Two-Sample Location Problem (2016, methodology): “Hence, the SS test is asymptotically as efficient as Srivastava et al.’s (2013) test in such settings.”
Optimal hypothesis testing for high dimensional covariance matrices (2012, background): “See also Srivastava [17].”
Testing mutual independence in high dimension via distance covariance (2016, background): “In recent years, extensive work has emerged in the high dimensional context, where p > n, including Ledoit and Wolf (2002), Jiang (2004), Schott (2005), Srivastava (2005, 2006), Chen et al. (2010), Fisher et al. (2010), Cai and Jiang (2011) and Fisher (2012) among others.”
On the sphericity test with large-dimensional observations (2013, background): “More recently, [26] examined the performance of T1 (a statistic first put forward in [24]) under non-normality, but with the moment condition γ = 3 + O(p), which essentially matches the Gaussian case (γ = 3) asymptotically.”
Asymptotic power of sphericity tests for high-dimensional data (2013, methodology): “Srivastava (2005) analyzes the asymptotic power of tests similar to those in Examples 1 and 2.”
On testing for high-dimensional white noise (2018, methodology): “Actually, Srivastava (2005) has derived the limiting distribution of the John’s test statistic with a general population covariance matrix and normally distributed samples.”
Corrections to LRT on large-dimensional covariance matrix by RMT (2009, methodology): “Related work include Ledoit and Wolf [6], Schott [8] and Srivastava [9].”
Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, background): “The results have been extended by Birke and Dette (2005); Srivastava (2005).”
Multivariate Theory for Analyzing High Dimensional Data (2007, background): “Tests are available in Srivastava (2005) to check if Σ = γ2I, γ2 unknown, or if Σ = diag(λ1 ,...,λ p) under the assumption that n = O(pδ), o Srivastava (2005) .”
Likelihood Ratio Tests for High‐Dimensional Normal Distributions (2013, background): “…size n. Readers are referred to the papers, for example, by Ledoit & Wolf (2002) and Chen et al. (2010) for the sphericity test; by Schott (2001, 2007) for testing the equality of multiple covariance matrices and by Srivastava (2005) for testing the covariance matrix of a normal distribution.”
Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing (2014, methodology): “[2, 3, 9, 14, 20, 21], so that LSS of the sample covariance matrices are approximated using either the CLT (1.”
New Weighted Portmanteau Statistics for Time Series Goodness of Fit Testing (2012, unknown): “Consider the problem discussed in Srivastava (2005) for testing for an identity covariance matrix.”