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The kernel estimate is relatively stable
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  • Published: December 1988

The kernel estimate is relatively stable

  • Luc Devroye1 

Probability Theory and Related Fields volume 77, pages 521–536 (1988)Cite this article

Summary

Consider the Parzen-Rosenblatt kernel estimate\(f_n = (1/n)\sum\limits_{i = 1}^n {K_h (x - X_i )}\) whereh>0 is a constant,K is an absolutely integrable function with integral one,K h (x)=(1/h), andX 1, ...,X n are iid random variables with common densityf onR d. We show that for all ε>0,

$$\mathop {\sup }\limits_{h > 0,f} P \leqq 2e^{ - \frac{{n\varepsilon ^2 }}{{32\mathop \smallint \limits^2 |K|.}}}$$

We also establish thatf n is relatively stable, i.e.

$$\frac{{\int {|f_n - f|} }}{{E\int | f_n - f|}} \to 1 in probability as n \to \infty ,$$

whenerver lim inf\(\sqrt n E\int | f_n - f| = \infty\). We also study what happens whenh is allowed to depend upon the data sequence.

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Author information

Authors and Affiliations

  1. School of Computer Science, McGill University, 805 Sherbrooke Street West, H3A 2K6, Montreal, Canada

    Luc Devroye

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  1. Luc Devroye
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Additional information

This work was sponsored by NSERC Grant A 3456 and FCAC Grant EQ-1678

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Devroye, L. The kernel estimate is relatively stable. Probab. Th. Rel. Fields 77, 521–536 (1988). https://doi.org/10.1007/BF00959615

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  • Received: 02 September 1986

  • Revised: 13 November 1987

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00959615

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Keywords

  • Data Sequence
  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Integrable Function
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