Open Access
September, 1982 Optimal Estimation of a General Regression Function
P. W. Millar
Ann. Statist. 10(3): 717-740 (September, 1982). DOI: 10.1214/aos/1176345867


Let $\Theta$ be a convex subset of $R^d$, and for each $\theta \in \Theta$ let $F(\theta; dt)$ be a probability on the line. For any vector $a_n = (a_{n1}, a_{n2}, \cdots, a_{nn})$ where $a_{ni} \in \Theta$, let $X_{n1}, \cdots, X_{nn}$ be independent observations, the distribution of $X_{ni}$ being $F(a_{ni}; dt)$. The main results give a method of estimating the unknown regression function $a_n$ based on a minimum distance recipe. Under regularity assumptions, the proposed estimators are shown, in an appropriate framework, to be asymptotically normal, locally asymptotically minimax, and robust. The abstract results are illustrated by application to the linear model and to exponential response models. In general, nothing at all is assumed about the form of the regression function; accordingly, this forces the limiting normal distributions of the proposed estimators to be located on infinite dimensional linear spaces.


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P. W. Millar. "Optimal Estimation of a General Regression Function." Ann. Statist. 10 (3) 717 - 740, September, 1982.


Published: September, 1982
First available in Project Euclid: 12 April 2007

zbMATH: 0488.62045
MathSciNet: MR663428
Digital Object Identifier: 10.1214/aos/1176345867

Primary: 62G05
Secondary: 62F12 , 62F35 , 62G20 , 62J02

Keywords: $\sqrt n$ consistency , Abstract Wiener space , asymptotic normality , Empirical distribution , exponetial family , Hilbert space , infinite dimensional normal families , Kiefer process , local asymptotic minimax , location model , minimum distance estimation , Non-parametric regression , robust regression

Rights: Copyright © 1982 Institute of Mathematical Statistics

Vol.10 • No. 3 • September, 1982
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