A universally acceptable smoothing factor for kernel density estimates

Abstract

We define a minimum distance estimate of the smoothing factor for kernel density estimates, based on a methodology first developed by Yatracos. It is shown that if fnh denotes the kernel density estimate on ℝd for an i.i.d. sample of size n drawn from an unknown density f, where h is the smoothing factor, and if fn is the kernel estimate with the same kernel and with the proposed new data-based smoothing factor, then, under a regularity condition on the kernel K, sup lim sup Ε∫|fn-f|dx/f n→∞ infh>0 Ε∫|fnh-f|dx ≤3. This is the first published smoothing factor that can be proven to have this property.