A note on the estimation of a distribution function and quantiles by a kernel method

There has recently been extensive work on the estimation by kernel methods of probability densities and their derivatives; for a review, see Wertz (1978). For estimating the cumulative distribution function, the empirical distribution function, F,, is in a sense already quite smooth, although it is known that further smoothing can be an advantage. The object of the present note is to report briefly an investigation of kernel estimation applied to distribution function, and quantiles. Further details are available from the author. For independent and identically distributed random variables X1, ...,X the usual kernel estimator of the density f (.) is