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A class of statistics g(v) is proposed for constructing tests and confidence intervals for a linear combination of means v using an incomplete multivariate normal sample. The class includes statistics based on maximum likelihood estimates for all the data, and simpler statistics such as those based on a subset of complete observations. A simple measure of relative efficiency is proposed to decide when a simpler statistic is appropriate. For the case of two variables with extra observations on one variable, approximate t distributions are suggested for some members of g, and resulting test statistics are compared in a simulation study.
Roderick J. A. Little (1976) studied this question.