ASYMPTOTIC PROPERTIES OF SMOOTHED vs. UNSMOOTHED CONDITIONAL DISTRIBUTION FUNCTION ESTIMATORS
K. L. Mehra, Yellela S.R. Krishnaiah, M. Sudhakara Rao · Bulletin of informatics and cybernetics · 1992
Let $ {(X_i, Y_i) : i = 1, 2, ldots } $ be a sequence of independent identically distributed random vectors in $ Re^2 $ with an absolutely continuous distribution, and let $ G_x(cdot) $ denote the conditional distribution function of $ Y_1 $ given $ X_1 = x $, 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 $ G_x(cdot) $ 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.