Asymptotic Performance Bounds for the Kernel Estimate

Abstract

We consider an arbitrary sequence of kernel density estimates $f_n$ with kernels $K_n$ possibly depending upon $n$. Under a mild restriction on the sequence $K_n$, we obtain inequalities of the type $E\big(\int|f_n - f|\big) \geq (1 + o(1))\Psi(n, f),$ where $f$ is the density being estimated and $\Psi(n, f)$ is a function of $n$ and $f$ only. The function $\psi$ can be considered as an indicator of the difficulty of estimating $f$ with any kernel estimate.