Abstract
This article deals with the problem of testing for two normal sub-mean vectors when the data set have two-step monotone missing observations. Under the assumptions that the population covariance matrices are equal, we obtain the likelihood ratio test (LRT) statistic. Furthermore, an asymptotic expansion for the null distribution of the LRT statistic is derived under the two-step monotone missing data by the perturbation method. Using the result, we propose two improved statistics with good chi-squared approximation. One is the modified LRT statistic by Bartlett correction,and the other is the modified LRT statistic using the modification coefficient by linear interpolation. The accuracy of the approximations are investigated by using a Monte Carlo simulation. The proposed methods are illustrated using an example.
Highlights
Standard statistical methods have been developed for analyzing complete rectangular data sets; incomplete data sets are often encountered
Under the null hypothesis in (1), the likelihood ratio test (LRT) statistic −2 log λ is asymptotically distributed as χ2 with p − p1 degrees of freedom when N (i) → ∞, i = 1, 2
The problems treated in this article concern making the modified LRT statistic based on the two-step monotone missing data
Summary
Standard statistical methods have been developed for analyzing complete rectangular data sets; incomplete data sets are often encountered. A test for sub-mean vectors with two-step monotone missing data under a onesample problem was discussed by Kawasaki and Seo (2016b). They derived the likelihood ratio (LR) criterion for testing the (p2 + p3)-mean vector under the given mean vector of p1-dimensions. They proposed an approximation of the upper percentile of the LRT statistic using linear interpolation based on Rao’s U statistic for complete data sets. The proof of a result is completed in the appendix
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