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Strong uniform consistency of nonparametric regression function estimates
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  • Published: August 1989

Strong uniform consistency of nonparametric regression function estimates

  • Hannelore Liero1 

Probability Theory and Related Fields volume 82, pages 587–614 (1989)Cite this article

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Summary

Let (X, Y) be a ℝdxℝ-valued random vector and let r(t)=E(Y/X=t) be the regression function of Y on X that has to be estimated from a sample (X i, Yi), i=1,..., n. We establish conditions ensuring that an estimate of the form

$$r_n (t) = {{\sum\limits_{i = 1}^n {Y_i } \Phi _{ni} (t,X_i )} \mathord{\left/ {\vphantom {{\sum\limits_{i = 1}^n {Y_i } \Phi _{ni} (t,X_i )} {\sum\limits_i^n {\Phi _{ni} (t,X_i )} }}} \right. \kern-\nulldelimiterspace} {\sum\limits_i^n {\Phi _{ni} (t,X_i )} }}$$

Where Φni(t, x) is a sequence of Borel measurable functions on ℝdxℝd, is uniformly strongly consistent with a certain rate of convergence. Applying this result we obtain rates of strong uniform consistency of the regressogram, kernel estimates, k n-nearest neighbor estimates and estimates based on orthogonal series.

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Authors and Affiliations

  1. Karl Weierstraß-Institut für Mathematik der Akademie der Wissenschaften der DDR, Mohrenstrasse 39, 1086, Berlin, German Democratic Republic

    Hannelore Liero

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  1. Hannelore Liero
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Liero, H. Strong uniform consistency of nonparametric regression function estimates. Probab. Th. Rel. Fields 82, 587–614 (1989). https://doi.org/10.1007/BF00341285

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  • Received: 28 January 1986

  • Revised: 01 October 1988

  • Issue Date: August 1989

  • DOI: https://doi.org/10.1007/BF00341285

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Keywords

  • Stochastic Process
  • Probability Theory
  • Measurable Function
  • Statistical Theory
  • Random Vector
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