A Note on the $L_1$ Consistency of Variable Kernel Estimates

Abstract

A sample $X_1, \cdots, X_n$ of i.i.d. $R^d$-valued random vectors with common density $f$ is used to construct the density estimate $f_n(x) = (1/n) \sum^n_{i = 1} H^{-d}_{ni}K((x - X_i)/H_{ni}),$ where $K$ is a given density on $R^d$, and the $H_{ni}$'s are positive functions of $n, i$ and $X_1, \cdots, X_n$ (but not of $x$). The $H_{ni}$'s can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of $\int |f_n - f|$ for all $f$. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).