Conditional empirical processes defined by φ - mixing sequences

Abstract

For a random vector (X, Y) in R 2 let m (y/x)=P(Y≤y|X=x) be the conditional distribution function. It is assumed that (X 1 , Y 1 ), (X 2 , Y 2 ), ... form a strictly stationary Φ-mixing sequence of random vectors with the same distribution as (X, Y). Considering an estimate mn (•|x) of m(•|x) it is shown that (nan) 1/2 {mn(•|x 0 )-m (•|x 0 )} converges to a certain Gaussian process depending on x 0 , where (an) is a sequence of bandwiths