The Annals of Statistics

Consistency for the least squares estimator in nonparametric regression

Marten Wegkamp and Sara van de Geer

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We shall study the general regression model $Y = g_0 (X) + \varepsilon$, where X and $varepsilon$ are independent. The available information about $g_0$ can be expressed by $g_0 \epsilon \mathscr{G}$ for some class $\mathscr{G}$. As an estimator of $g_0$ we choose the least squares estimator. We shall give necessary and sufficient conditions for consistency of this estimator in terms of (basically) geometric properties of $\mathscr{G}$. Our main tool will be the theory of empirical processes.

Article information

Ann. Statist., Volume 24, Number 6 (1996), 2513-2523.

First available in Project Euclid: 16 September 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 62J02: General nonlinear regression

Consistency empirical process entropy Glivenko-Cantelli classes least squares estimation regression


van de Geer, Sara; Wegkamp, Marten. Consistency for the least squares estimator in nonparametric regression. Ann. Statist. 24 (1996), no. 6, 2513--2523. doi:10.1214/aos/1032181165.

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