The Annals of Statistics

Local Extremes, Runs, Strings and Multiresolution

P. L. Davies and A. Kovac

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The paper considers the problem of nonparametric regression with emphasis on controlling the number of local extremes. Two methods, the run method and the taut-string multiresolution method, are introduced and analyzed on standard test beds. It is shown that the number and locations of local extreme values are consistently estimated. Rates of convergence are proved for both methods. The run method converges slowly but can withstand blocks as well as a high proportion of isolated outliers. The rate of convergence of the taut-string multiresolution method is almost optimal. The method is extremely sensitive and can detect very low power peaks.

Section 1 contains an introduction with special reference to the number of local extreme values. The run method is described in Section 2 and the taut-string-multiresolution method in Section 3. Low power peaks are considered in Section 4. Section contains a comparison with other methods and Section 6 a short conclusion. The proofs are given in Section 7 and the taut-string algorithm is described in the Appendix.

Article information

Ann. Statist., Volume 29, Number 1 (2001), 1-65.

First available in Project Euclid: 5 August 2001

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 65D10: Smoothing, curve fitting 62G20: Asymptotic properties

Nonparametric regression local extremes runs strings multiresolution analysis asymptotics outliers low power peaks


Davies, P. L.; Kovac, A. Local Extremes, Runs, Strings and Multiresolution. Ann. Statist. 29 (2001), no. 1, 1--65. doi:10.1214/aos/996986501.

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