ASYMPTOTICS OF THE RESIDUALS DENSITY ESTIMATION IN NONPARAMETRIC REGRESSION UNDER m(n)-DEPENDENT SAMPLE

摘要

Let Y_i = M(X_i) + e_i, where M(x) = E(Y | X = x) is an unknown real function on B(is contained in R), {(X_i,Y_i)} is a stationary and m(n)-dependent sample from (X, Y), the residuals {e_i} are independent of {X_i} and have unknown common density f(x). In [2] a nonparametric estimate f_n(x) for f(x) has been proposed on the basis of the residuals estimates. In this paper, we further obtain the asymptotic normality and the law of the iterated logarithm of f_n(x) under some suitable conditions. These results together with those in [2] bring the asymptotic theory for the residuals density estimate in nonparametric regression under m(n)-dependent sample to completion.