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Estimating the direction in which a data set is most interesting
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  • Published: December 1988

Estimating the direction in which a data set is most interesting

  • Peter Hall1 

Probability Theory and Related Fields volume 80, pages 51–77 (1988)Cite this article

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  • 15 Citations

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Summary

Estimation of orientation is a key operation at each step in projection pursuit. Since projection pursuit is a nonparametric algorithm, and since even low-dimensional approximations to the target function must converge to their limits at rates considerably slower than n 2-1 (where n is sample size), then it might be thought that the same is true of orientation estimates. It is shown in the present paper that this is not the case, and that estimation of orientation is a parametric operation, in the sense that, under mild nonparametric assumptions, correctly-chosen kernel-type orientation estimates converge to their limits at rate n 2-1 . This property is not enjoyed by standard projection pursuit orientation estimates, which converge at a slower rate than n 2-1 . Most attention in the present paper is focussed on the case of projection pursuit density approximation, but it is pointed out that our arguments hold generally. An important practical conclusion is that data should be smoothed less when estimating orientation than when constructing the final projection pursuit approximation.

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Authors and Affiliations

  1. Department of Statistics, Australian National University, GPO Box 4, 2601, Canberra, ACT, Australia

    Peter Hall

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  1. Peter Hall
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Hall, P. Estimating the direction in which a data set is most interesting. Probab. Th. Rel. Fields 80, 51–77 (1988). https://doi.org/10.1007/BF00348752

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  • Received: 21 May 1987

  • Revised: 02 May 1988

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00348752

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Keywords

  • Stochastic Process
  • Probability Theory
  • Slow Rate
  • Statistical Theory
  • Density Approximation
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