### A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting

high-dimensional two-sample testing problem

Ahmad, Muhammad Rauf (2012). A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting.
[Preprint] (Submitted)

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## Abstract

A two-sample test statistic is presented for testing the equality of multivariate mean vectors when the dimension of the vectors, $p$, exceeds the sample sizes, $n_i$, $i = 1, 2$, and the underlying distributions are not necessarily normal. Under mild assumptions on the traces of the covariance matrices of the two independent samples, the test statistic is shown to be asymptotically chi-square distributed when both $p \rightarrow \infty$ and $n_i \rightarrow \infty$. However, the validity of the test statistic when $p$ is fixed but large, including $p > n_i$, and when the data follow multivariate normal distribution, is shown as special cases of the main result. This two-sample chi-square approximation eventually helps us establish the validity of Box's approximation for high-dimensional and non-normal data to a two-sample set up, valid even under Behrens-Fisher setting. The limiting chi-square distribution of the statistic is obtained using the asymptotic theory of degenerate $U$-statistics, and using a result from classical asymptotic theory, the asymptotic limit is further extended to an approximate normal distribution. Both independent and paired sample cases are considered.

Authors/Creators: Ahmad, Muhammad Rauf A U-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens-Fisher setting high-dimensional two-sample testing problem Annals of the Institute of Statistical Mathematics (0020-3157) 2012 31 0020-3157 English Additional affiliation of the author: Department of Statistics, Uppsala University, Sweden. The full text of this publication is not available for reading until the 28:th of February 2013. Preprint Submitted version None U Auxiliary disciplines > U10 Mathematical and statistical methods (A) Swedish standard research categories 2011 > 1 Natural sciences > 101 Mathematics > 10106 Probability Theory and Statistics High-dimensional multivariate inference , Box's approximation, Behrens-Fisher setting, Degenerate U-statistics urn:nbn:se:slu:epsilon-e-589 http://urn.kb.se/resolve?urn=urn:nbn:se:slu:epsilon-e-589 9001 (NL, NJ) > Dept. of Energy and Technology Mr. M. Rauf Ahmad 28 Aug 2012 07:40 02 Dec 2014 10:51

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