Two-sample behrens-fisher problem for high-dimensional data

Abstract

This article is concerned with the two-sample Behrens-Fisher problem in high-dimensional settings. A novel test is proposed that is scale-invariant, asymp- totically normal under certain mild conditions, and the dimensionality is allowed to grow in the rate, respectively, from square to cube of the sample size in different sce- narios. We explain the necessity of bias correction for existing scale-invariant tests, otherwise they do not have well-defined limits even under the null hypothesis. We also give some examples to theoretically show the advantage of the scale-invariant test over scale-variant tests when variances of the two samples are different. This article is concerned with the two-sample Behrens-Fisher problem in high-dimensional settings. Assume that {Xi1; · · · ;Xini } for i = 1; 2 are two independent random samples with the sizes n1 and n2, from p-variate distribu- tions F (x − 1) and G(x − 2) located at p-variate centers 1 and 2. Denote n = n1 + n2. We wish to test is developed under the equality of the two covariances, say 1 = 2 = . The key feature of the Bai and Saranadasa's proposal is to use the Euclidian norm to replace the Mahalanobis norm since having the inverse of the sample covariance matrix is no longer beneficial when p=n → c > 0. Zhang and Xu (2009) extended this method to the k-sample high-dimensional Behrens-Fisher problem and derived the asymptotic distribution of the test statistic when p=n → c < 1.