On efficient estimation in continuous models based on finitely quantized observations

Abstract

We consider minimum phi-divergence estimators (theta) over cap (phi)(n) of parameters theta of arbitrary dominated models mu(theta) << lambda on the real line, based on finite quantizations of i.i.d. observations X-1,..., X-n from these models. The quantizations are represented by finite interval partitions P-n = (A(n1),...,A(nmn)) of the real line, where m(n) is allowed to increase to infinity for n --> infinity. The models with densities f(theta) = d mu(theta)/d lambda are assumed to be regular in the sense that they admit finite Fisher informations J(theta). In the first place we have in mind continuous models dominated by the Lebesgue measure lambda. Owing to the quantizations, (theta) over cap (phi)(n) are discrete-model estimators for which the desirable properties ( computation complexity, robustness, etc.) can be controlled by a suitable choice of functions phi. We formulate conditions under which these estimators are consistent and efficient in the original models mu(theta) in the sense that root n((theta) over cap (phi)(n)-theta) -->(L) N(0, J(theta)(-1)) as n --> infinity.