Efficient estimates and optimum inference procedures in large samples

Abstract

The concept of efficiency in estimation is linked with closeness of approximation to the derivative of log likelihood, which plays an important role in statistical inference in large samples. Various orders of efficiency are defined depending on degrees of closeness, and properties of estimates satisfying these criteria are studied. Such measures of efficiency appear to be more appropriate than the one related to asymptotic variance of an estimate for judging the performance of an estimate, when used as a substitute for the whole sample in drawing inference about unknown parameters. It is found that, under some conditions, the maximum likelihood estimate has some optimum properties which distinguish it from all other large sample estimates.