A characterization of limiting distributions of regular estimates

Abstract

We consider a sequence of estimates in a sequence of general estimation problems with a k-dimensional parameter. Under certain very general conditions we prove that the limiting distribution of the estimates, if properly normed, is a convolution of a certain normal distribution, which depends only of the underlying distributions, and of a further distribution, which depends on the choice of the estimate. As corollaries we obtain inequalities for asymptotic variances and for asymptotic probabilities of certain sets, generalizing so some results of J. Wolfowitz (1965), S. Kaufman (1966), L. Schmetterer (1966) and G. G. Roussas (1968). © 1970 Springer-Verlag.