A Note on the L₁ Consistency of Variable Kernel Estimates

A sample X₁, ⋯, Xₙ of i.i.d. Rᵈ-valued random vectors with common density f is used to construct the density estimate fₙ(x) = (1/n) ∑ⁿi = 1 H⁻ᵈₙᵢK((x - Xᵢ)/Hₙᵢ), where K is a given density on Rᵈ, and the Hₙᵢ's are positive functions of $n, i$ and X₁, ⋯, Xₙ (but not of x). The Hₙᵢ's can be thought of as locally adapted smoothing parameters. We give sufficient conditions for the weak convergence to 0 of ∫ |fₙ - f| for all f. This is illustrated for the estimate of Breiman, Meisel and Purcell (1977).