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

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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.

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Acknowledgments

The author is sincerely thankful to Prof. Dr. Anne Leucht, Theoretical Econometrics and Statistics, Department of Economics, University of Mannheim, Germany, for her careful perusal of the manuscript and helpful suggestions. Thanks are also due to the editor, the associate editor, and to two anonymous referees for their helpful suggestions and comments which lead to this improved version of the article.

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Authors and Affiliations

  1. Department of Statistics, Uppsala University, Ekonomikum, Kyrkogårdsgatan 10, 75120 , Uppsala, Sweden

    M. Rauf Ahmad

  2. Department of Energy and Technology, Swedish University of Agricultural Sciences, Gerda Nilssons väg, 5, 75651 , Uppsala, Sweden

    M. Rauf Ahmad

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Ahmad, M.R. A \(U\)-statistic approach for a high-dimensional two-sample mean testing problem under non-normality and Behrens–Fisher setting. Ann Inst Stat Math 66, 33–61 (2014). https://doi.org/10.1007/s10463-013-0404-2

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  • DOI: https://doi.org/10.1007/s10463-013-0404-2

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