This paper introduces a new approach to prediction by bringing together two different nonparametric ideas: distribution free inference and nonparametric smoothing. Specifically, we consider the problem of constructing nonparametric tolerance/prediction sets. We start from the general conformal prediction approach and we use a kernel density estimator as a measure of agreement between a sample point and the underlying distribution. The resulting prediction set is shown to be closely related to plug-in density level sets with carefully chosen cut-off values. Under standard smoothness conditions, we get an asymptotic efficiency result that is near optimal for a wide range of function classes. But the coverage is guaranteed whether or not the smoothness conditions hold and regardless of the sample size. The performance of our method is investigated through simulation studies and illustrated in a real data example.
Keywords: conformal prediction; distribution free; finite sample; kernel density; prediction sets.