On the Likelihood Ratio Test for the Equality of Multivariate Normal Populations with Two-step Monotone Missing Data

In this paper, we consider the problem of testing the equality of multivariate normal populations when the data set has missing observations with a two-step monotone pattern. The likelihood ratio test (LRT) statistic for the simultaneous testing of the mean vectors and the covariance matrices is given under the condition of two-step monotone missing data. An approximate modified likelihood ratio test (MLRT) statistic is presented using linear interpolation based on the coefficients of the MLRT statistic in the case of complete data sets. As an alternative approach, we propose approximate MLRT statistics of two kinds with two-step monotone missing data using the decompositions of the likelihood ratio (LR). An approximate upper percentile of the LRT statistic with two-step monotone missing data is also derived based on an asymptotic expansion for the LRT statistic in the case of complete data sets. Finally, we investigate the accuracy of the approximations using Monte Carlo simulation. AMS 2010 Mathematics Subject Classification. 62E20, 62H10.