WIAS Preprint No. 154, (1995)

On the estimation of a support curve of indeterminate sharpness



Authors

  • Hall, Peter
  • Nussbaum, Michael
  • Stern, Steven E.

2010 Mathematics Subject Classification

  • 62G07 62H05 62H10

Keywords

  • Convergence rate, curve estimation, endpoint, order statistic, regular variation, support

Abstract

We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focussed on cases where the rate of decrease is relatively fast this, being the most difficult setting. It demands the use of a relatively large number of bivariate order statistics. By way of comparison, support curve estimation in the context of slow rates of decrease of the density may be addressed using methods that use only a relatively small number of order statistics at the extremities of the point cloud. In this paper we suggest a new type of estimator, based on projecting onto an axis those data values lying within a thin rectangular strip. Adaptive univariate methods are then applied to the problem of estimating an endpoint of the distribution on the axis. The new method is shown to have theoretically optimal performance in a range of settings. Its numerical properties are explored in a simulation study.

Appeared in

  • J. Multivariate Anal. 62 (1997) no. 2 pp. 204-232

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