ASYMPTOTIC PROPERTIES OF SMOOTHED vs. UNSMOOTHED CONDITIONAL DISTRIBUTION FUNCTION ESTIMATORS

Let {(Xi, Y1) : i = 1, 2, ... } be a sequence of independent identically distributed random vectors in 12 with an absolutely continuous distribution, and let G.J.) denote the conditional distribution function of Y1 given X1 = .v, assuming that it exists. In this paper, the asymptotic normality and almost sure convergence rates for smoothed rank nearest neighbor and Nadaraya Watson type estimators of G1(•) are established. It is also shown. using the concept of deficiency, that smoothed estimators are superior (asymptotically) to the corresponding unsmoothed ones under appropriate choice of the smoothing kernels.