KERNEL ESTIMATION OF THE DENSITY OF A STATISTIC

Let X(1),...,X(n) be a stationary sequence of random variables with common density f(.) and let t(n):= t(n)(X(1),...,X(n)) be a statistic. The problem of estimating f(.) has often been addressed, particularly in the i.i.d. setup. Using the kernel method, we address the problem of estimating the density of the statistic t(n), for both i.i.d. data and observations from a stationary time series. We present estimators for both scenarios and prove their consistency (in mean square) under mild assumptions. In the i.i.d. setup, the proposed estimators can be used to smooth replicates obtained from the grouped jackknife or the bootstrap. Also, bandwidth choice is discussed and rates of convergence are given.