A PRINCIPAL COMPONENTS APPROACH TO SOME MULTIVARIATE ESTIMATION PROBLEMS WITH INCOMPLETE DATA

The maximum likelihood estimators of the parameters of a multivariate normal distribution based on monotone samples and their first two moments were derived in the literature. When multicollinearity is present, these estimators are unreliable. By using a subset of the sample covariance matrix's principal components we construct estimators free of the distortions caused by multicollinearity. The new estimators are shown to be consistent and asymptotically unbiased. We derive the distributions of various functions of the sample covariance matrix pertaining to those estimators and obtain closed form expressions for the first two moments of the estimators based on principal components. These expressions, however, are not easily evaluated. We also examine the multivariate regression problem in monotone samples. When there is no indication of multicollinearity, we propose a set of estimators for the regression parameters which are shown to be consistent, asymptotically unbiased and to have smaller variances than the usual least squares regression estimators. Principal component regression is often applied in cases of highly correlated "independent" variables. The technique is extended in monotone multicollinear samples. The estimators produced have some optimum properties. Results related to the development of the principal components' estimators and examples are also considered.