Two-sample Tests of Sub-mean Vectors Under Two-step Monotone Missing Data

This study proposes a novel test statistic for the two-sample problem involving sub-mean vectors under a two-step monotone missing data structure. The proposed procedure is constructed based on the structure of Rao's U-statistic by combining a Hotelling's T^2-type statistic for monotone missing data with the standard Hotelling's T^2 statistic, thereby efficiently utilizing the available information in incomplete observations. We consider the problem of testing the equality of sub-mean vectors between two populations under the assumption that a subset of the mean components is common. The asymptotic expansion of the null distribution of the proposed statistic is derived, and its distribution function and approximate upper percentiles are obtained. To improve the accuracy of the chi-squared approximation in finite samples, Bartlett and Bartlett-type correction methods are also developed. The performance of the proposed approximations and correction procedures is investigated through extensive Monte Carlo simulations under various dimensional and sample size settings. A numerical example based on real data is presented to illustrate the applicability and practical usefulness of the proposed methodology.