Asymptotics of Conditional Empirical Processes

We investigate asymptotic approximations of two estimators of the conditional empirical process of Y given X= x, leading to a functional law of the iterated logarithm. Our main results show that the weak behavior of the conditional empirical processes at a given x is essentially the same as that of the empirical process, although the rates now depend on the bandwidth. We examine the kernel estimator of Nadaraya [20] and Watson [29] and the nearest neighbor (NN) estimator of Yang [31] and others. The subject of nonparametric regression was reviewed by Stone [24,25] and Collomb [4]. Recently Rev&z [22], Johnston [12], Liero [17], and Konakov and Piterbarg [ 151 developed strong approximations of residual mean regression functions. Mack and Silverman [18] and Cheng [2]