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

M. Ahmad ()

Annals of the Institute of Statistical Mathematics, 2014, vol. 66, issue 1, 33-61

Abstract: A two-sample test statistic is presented for testing the equality of mean vectors when the dimension, $$p$$ , exceeds the sample sizes, $$n_i,\; i=1, 2$$ , and the distributions are not necessarily normal. Under mild assumptions on the traces of the covariance matrices, the statistic is shown to be asymptotically Chi-square distributed when $$n_i, p \rightarrow \infty $$ . However, the validity of the test statistic when $$p$$ is fixed but large, including $$p > n_i$$ , and when the distributions are multivariate normal, is shown as special cases. This two-sample Chi-square approximation helps us establish the validity of Box’s approximation for high-dimensional and non-normal data to a two-sample setup, 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, it is further extended to an approximate normal distribution. Both independent and paired-sample cases are considered. Copyright The Institute of Statistical Mathematics, Tokyo 2014

Keywords: High-dimensional multivariate inference; Box’s approximation; Behrens–Fisher setting; Degenerate $$U$$ -statistics (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (6)

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DOI: 10.1007/s10463-013-0404-2

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