Simultaneous Tests for Mean Vectors and Covariance Matrices with Three-Step Monotone Missing Data

In this paper, we consider simultaneous tests of the mean vectors and the covariance matrices under three-step monotone missing data for a one-sample and a multi-sample problem. We provide the likelihood ratio test (LRT) statistic and propose statistics for improving the accuracy of the χ² approximation. These test statistics are derived by decomposing the likelihood ratio (LR) using the coefficients of the modified LRT statistics with complete data. As an alternative approach, we derive an approximate upper percentile of the LRT statistic with three-step monotone missing data using linear interpolation based on an asymptotic expansion of the LRT statistic with complete data. Finally, we investigate the asymptotic behavior of the upper percentiles of these test statistics and the accuracy of approximate upper percentiles via Monte Carlo simulation. In addition, we give an example of test statistics and approximate upper percentiles proposed in this paper.