Mean Test with Fewer Observation than Dimension and Ratio Unbiased Estimator for Correlation Matrix

Abstract

Hotelling's T-squared test is a classical tool to test if the normal mean of a multivariate normal distribution is a specified one or the means of two multivariate normal means are equal. When the population dimension is higher than the sample size, the test is no longer applicable. Under this situation, in this paper we revisit the tests proposed by Srivastava and Du (2008), who revise the Hotelling's statistics by replacing Wishart matrices with their diagonal matrices. They show the revised statistics are asymptotically normal. We use the random matrix theory to examine their statistics again and find that their discovery is just part of the big picture. In fact, we prove that their statistics, decided by the Euclidean norm of the population correlation matrix, can go to normal, mixing chi-squared distributions and a convolution of both.Examples are provided to show the phase transition phenomenon between the normal and mixing chi-squared distributions. The second contribution of ours is a rigorous derivation of an asymptotic ratio-unbiased-estimator of the squared Euclidean norm of the correlation matrix.

Asymptotic Independence of the Sum and Maximum of Dependent Random Variables with Applications to High-Dimensional Tests

“…Part (i) of Theorem S6 is from Srivastava and Du (2008). Recently Jiang and Li (2021) obtained a general theory, which also leads to the same conclusion. Same as in the onesample test, for test T…”

“…Part (i) of the above theorem is from Srivastava (2009), which is also a corollary of the recent work by Jiang and Li (2021). For the sum-type test, a level-α test will be performed through rejecting H 0 when T…”

“…There exists a constant C > 0 such that P (χ 2 (n − 1) < 1 2 n) ≤ e −Cn for all n ≥ 1 and Finally we prove part (iii). According to the proof of Theorem 3.1 in Srivastava (2009) or the proof of Theorem 1 in Jiang and Li (2021), we have…”

“…According to Section 5 and the proof of Theorem 2.1 in Srivastava and Du (2008) or the proof of Theorem 2 in Jiang and Li (2021), we have…”

A Pairwise Hotelling Method for Testing High-Dimensional Mean Vectors

“…Feng et al (2017) assumed a block diagonal structure for the covariance matrix, and within each block, Hotelling's T 2 can be applied. Further studies on DHT include, for example, Srivastava (2009), Park and Ayyala (2013), Srivastava et al (2013), Feng and Sun (2015), , Gregory et al (2015), Dong et al (2016), Cao et al (2018), Chen et al (2019), and Jiang and Li (2021).…”