In this study, we examine the problem of testing for the sub-mean vector when the data have a two-step monotone missing pattern. In particular, when the data set consists of complete data with \(p (=p_1+p_2+p_3+p_4)\) dimensions and incomplete data with \(p_1+p_2\) dimensions, we consider the one-sample problem of testing the \(p_2+p_3+p_4\) mean vector, \(p_3+p_4\) mean vector, and \(p_4\) mean vector under the provided mean vector of remaining dimensions. Considering the hypothesis of a sub-mean vector with two-step monotone missing data, we construct a test statistic based on Rao’s U-statistic structure. In addition, we derive the stochastic expansion of these test statistics for the case where the sample size is large. We present the upper percentiles for the null distribution. Furthermore, we propose a Bartlett-corrected transformation statistic as well as investigate the accuracy using Monte Carlo simulation. Finally, we demonstrate the usefulness of the approximate upper percentiles of the test statistic through numerical example test.
