Strong consistency of automatic kernel regression estimates

Regression function estimation from independent and identically distributed bounded data is considered. TheL2 error with integration with respect to the design measure is used as an error criterion. It is shown that the kernel regression estimate with an arbitrary random bandwidth is weakly and strongly consistent forall distributions whenever the random bandwidth is chosen from some deterministic interval whose upper and lower bounds satisfy the usual conditions used to prove consistency of the kernel estimate for deterministic bandwidths. Choosing discrete bandwidths by cross-validation allows to weaken the conditions on the bandwidths.