## The Annals of Statistics

### Optimal Plug-in Estimators for Nonparametric Functional Estimation

#### Abstract

Consider the problem of estimating the value of a functional $\Lambda(f)$ for $f$ an unknown density or regression function. The straightforward plug-in estimator $\Lambda(\hat f)$ with $\hat f$ a particular estimate of $f$ achieves the optimal rate of convergence in the sense of Stone over bounded subsets of a Sobolev space for a broad class of linear and nonlinear functionals. For many functionals the rate calculation depends on a Frechet-like derivative of the functional, which may be obtained using elementary calculus. For some classes of functionals, $\hat f$ is undersmoothed relative to what would be used to estimate $f$ optimally. Examples for which a plug-in estimator is optimal include $L^q$ norms of regression or density functions and their derivatives and the expected integrated squared bias. When interested in computing estimates over classes of functions which satisfy certain restrictions, such as strict positivity or boundary conditions, the plug-in estimator may or may not be optimal, depending on the functional and the function class. The functional calculus establishes conditions under which the plug-in estimator remains optimal, and sometimes suggests an appropriate modification when it does not.

#### Article information

Source
Ann. Statist., Volume 20, Number 3 (1992), 1306-1328.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176348770

Digital Object Identifier
doi:10.1214/aos/1176348770

Mathematical Reviews number (MathSciNet)
MR1186251

Zentralblatt MATH identifier
0763.62023

JSTOR