Estimating Multiple Derivatives Simultaneously: What Is Optimal?

[摘要] Nonparametric regression techniques including kernel smoothing [1], spline smoothing [2], and local regression [3] are useful for estimating a mean response function µ(x) in the statistical model Yi = µ (xi)+?i when one is unwilling to assume that µ(x) is linear (or polynomial of higher but known degree) in the covariate x. These same techniques can also be employed to estimate one or more derivatives of µ(x). While the techniques differ in their details, they have a common underlying theme. One specifies a covariate value x0 and estimates µ(x) or one of its derivatives at x0 by solving an optimization problem that is localized to a neighborhood of x0, in that only observations with covariate values inside the neighborhood contribute substantively to the solution. For example, the simplest incarnation of this theme is to define µ(x0) to be the average of all responses Yi for which |xi-x0| is sufficiently small. As one slides x0 through a continuum of all possible covariate values, an estimated mean response or derivative is then traced out. Selecting the neighborhood size is a crucial implementation decision to which much literature has been devoted.

[关键词] Multiple Derivatives;Simultaneously;Optimal