A test for the mean vector with fewer observations than the dimension

Abstract

In this paper, we consider a test for the mean vector of independent and identically distributed multivariate normal random vectors where the dimension p is larger than or equal to the number of observations N. This test is invariant under scalar transformations of each component of the random vector. Theories and simulation results show that the proposed test is superior to other two tests available in the literature. Interest in such significance test for high-dimensional data is motivated by DNA microarrays. However, the methodology is valid for any application which involves high-dimensional data.

A review of 20 years of naive tests of significance for high-dimensional mean vectors and covariance matrices

“…While acknowledging the defect of the Hotelling test, as indicated in [3,16], Srivastava and Du (2008) [42] noted that the NET and ANT are not scale invariant, which may cause lower power when the scales of different components of the model are very different. Accordingly, they proved the following modification to the ANT:…”

“…where λ(R) is the largest eigenvalue of the correlation matrix R. Srivastava and Du [42] showed that if n ≍ p η and 1 2 < η 1,…”

“…Srivastava and Du also considered the two-sample location problem with the common covariance matrix Σ Σ Σ [42] and proposed the testing statistic…”

Test for the mean matrix in a Growth Curve model for high dimensions

“…and c q,n = 1 + trR 2 /q 3/2 is an adjustment factor converging to 1 in probability as (n, q) → ∞, n = O(q δ ), δ > 1/2 proposed by Srivastava and Du (2008). Define the population correlation matrix as…”

“…Srivastava (2007) proposed a Hotelling's T 2 type test, by using Moore-Penrose inverse of the sample covariance matrix instead of the inverse when N is smaller than p. It may be noted that all the above discussed tests are invariant under the group of orthogonal matrices. A test that is invariant under the group of non-singular diagonal matrices has recently been proposed by Srivastava and Du (2008) under the normal distribution and Srivastava (2009) under non-normality. It may be noted that this test is not invariant under the transformation by orthogonal matrices.…”