On the estimation of a support curve of indeterminate sharpness

Peter Hall*, Michael Nussbaum, Steven E. Stern

*Corresponding author for this work

Research output: Contribution to journalArticleResearchpeer-review

29 Citations (Scopus)


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 focused 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 employ 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.

Original languageEnglish
Pages (from-to)204-232
Number of pages29
JournalJournal of Multivariate Analysis
Issue number2
Publication statusPublished - Aug 1997
Externally publishedYes


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