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Efromovich, Sam

On nonparametric regression for iid observations in a general setting. (English) Zbl 0865.62025

Ann. Stat. 24, No. 3, 1126-1144 (1996).
Summary: 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.

Cited in 13 Documents

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62K05 Optimal statistical designs
62L12 Sequential estimation

Keywords:

curve estimation; sharp-optimal estimation; random design nonparametric regression; response function; optimal minimax mean integrated squared error convergence; best constant; rate of risk convergence; nonparametric analog of Fisher information; location; scale; censored data; mixture models
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References:

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