On pointwise laws of iterated logarithm for estimators of certain conditional functionals

Abstract

We study the estimation of certain functionals of the conditional distribution function with two classes of nonparametric estimators—the rank nearest neighbour (RNN) type estimators and the Nadaraya-Watson (NW) kernel type estimators. We obtain sharp pointwise rates of strong consistency by establishing laws of the iterated logarithm for these two classes of estimators. The results parallel those of Hall [Hall, P., 1981, Laws of the iterated logarithm for nonparametric density estimators. Zeitschrift Fur Wahrscheinlichkeitstheorie Und Verwandte Gebiete, 56, 47–61] and Härdle [Härdle, W., 1984, A law of the iterated logarithm for nonparametric regression function estimators. The Annals of Statistics, 12, 624–635] for certain density and regression function estimators respectively, and extend those of Mehra et al. [Mehra, K.L., Rama Krishnaiah, Y.S. and Rao, S.M., 1992a, Asymptotic properties of smoothed vs. unsmoothed conditional distribution function estimators, Bulletin of Informatics and Cybernetics, 25, 71–97] on the strong consistency of smooth conditional distribution function estimators.