The Annals of Statistics

On nonparametric regression for IID observations in a general setting

Sam Efromovich

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We consider the problem of sharp-optimal estimation of a response function $f(x)$ in a random design nonparametric regression under a general model where a pair of observations $(Y, X)$ has a joint density $p(y, x) = p(y|f(x)) \pi(x)$. We wish to estimate the response function with optimal minimax mean integrated squared error convergence as the sample size tends to $\infty$. Traditional regularity assumptions on the conditional density $p(y| \theta)$ assumed for parameter $\theta$ estimation are sufficient for sharp-optimal nonparametric risk convergence as well as for the existence of the best constant and rate of risk convergence. This best constant is a nonparametric analog of Fisher information. Many examples are sketched including location and scale families, censored data, mixture models and some well-known applied examples. A sequential approach and some aspects of experimental design are considered as well.

Article information

Ann. Statist., Volume 24, Number 3 (1996), 1126-1144.

First available in Project Euclid: 20 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 62J02: General nonlinear regression 62E20: Asymptotic distribution theory 62F35: Robustness and adaptive procedures

Nonparametric regression curves estimation sharp-optimal risk convergence sequential estimation location and scale families censored data mixtures


Efromovich, Sam. On nonparametric regression for IID observations in a general setting. Ann. Statist. 24 (1996), no. 3, 1126--1144. doi:10.1214/aos/1032526960.

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