An Application of the Efron-Stein Inequality in Density Estimation

Abstract

The Efron-Stein inequality is applied to prove that the kernel density estimate f_nf_n, with an arbitrary nonnegative kernel and an arbitrary smoothing factor, satisfies the inequality \operatorname{var}(\int|f_n - f|) \leq 4/n\operatorname{var}(\int|f_n - f|) \leq 4/n for all densities ff. Similar inequalities are obtained for other estimates.