Empirical Plug-in Curve and Surface Estimates

We obtain estimates of a surface θ : Rd → R by first finding the coefficient vector β(x) that minimizes a distance between θ(z) and an approximating function θ(z−x;β) for z in a neighborhood of a given point x ∈ Rd, next expressing this β(x) as a functional β(x;G) of a surface G : Rq → R that admits an empirical estimate Ĝ(·), and then using the empirical plug-in approach with β̂(x) = β(x; Ĝ) and θ̂(x) = θ(0, β̂(x)). Examples of G’s are the multivariate distribution function and the integrated hazard function. With this approach the random part of the estimation error can be expressed in terms of empirical processes of the form v(·,x)d[Ĝ(·) − G(·)], which facilitates asymptotic analysis. This approach gives simple derivations of old and new curve and surface estimates as well as their properties. It is particularly useful for providing simple and highly efficient estimates in boundary regions. We apply the approach to density estimation, univariate and multivariate; to hazard estimation; and to nonparametric regression.