An investigation of the rank transform in the multivariate one-sample location problem

In the multivariate one-sample location problem, the size of Hotelling's T² test fluctuates widely from nominal alpha as the distributions deviate from multinormality due to skewness or the distributions have tails heavier than multinormal distributions. For skewed distributions, several modifications of Hotelling's T² are available to adjust for skewness. In this paper for symmetric distributions, a test statistic based on the signed ranks of the data is discussed and investigated. The proposed test statistic is obtained by computing Hotelling's T² statistic as a function of the signed ranks and is equivalent to a statistic of Puri and Sen (1971). An approximation to its null distribution is derived by moment matching criterion. Monte Carlo results are presented to compare several test statistics. For various multidimensional distributions the Monte Carlo results indicate the robustness of the proposed statistics. For bivariate normal, uniform and Cauchy distributions, the results of a power comparison study are summarized.