Simultaneous testing of the mean vector and the covariance matrix with two-step monotone missing data

In this paper, we consider the problem of simultaneous testing of the mean vector and the covariance matrix when the data have a two-step monotone pattern that is missing observations.We give the likelihood ratio test (LRT) statistic and propose an approximate upper percentile of the null distribution using linear interpolation based on an asymptotic expansion of the modified LRT statistic in the case of a complete data set.As another approach, we give the modified LRT statistics with a two-step monotone missing data pattern using the coefficient of the modified LRT statistic with complete data.Finally, we investigate the asymptotic behavior of the upper percentiles of these test statistics by Monte Carlo simulation.