Estimation of Parameters from Incomplete Data

THE present note is concerned with a special case of the general problem of obtaining efficient estimators for the parameters of a normal multivariate population when the available sample data are incomplete in the sense that measures on all variables are not available for all individuals in the sample. Such fragmentary data may arise because part of the data are irretrievably lost (e.g., in an archaeological find), or because certain data were purposely not collected. The decision not to measure all individuals in the sample on all variables may be reached because of the cost of measurement, because of limited time, because the measurement of a certain variable alters or destroys the individual measured (e.g., in mental testing, testing explosives), and so forth. The general problem for normal bivariate populations has been treated by Wilks 3. This treatment has been further generalized to normal multivariate populations by Matthai 2, who deals explicitly with the trivariate case. Unfortunately, the general maximum likelihood equations have proved rather intractable, and no simple formulas for the maximum likelihood estimators are available in the general case, even for a sample from a bivariate population. In the present paper, the problem of estimating the parameters of a normal trivariate population from incomplete data is dealt with in a special case for which explicit solutions to the maximum likelihood equations are readily obtained. This special case is described in the following section. Formulas for the maximum likelihood estimators are given; their application is illustrated by a numerical example. The sampling variances and covariances of the maximum likelihood estimators are derived. An examination is made of the efficiency of the usual methods that utilize only that portion of the data that is complete. The foregoing results are specialized to apply to a commonly encountered bivariate (rather than trivariate) situation.