A note on estimation of the shape of density level sets of star-shaped distributions

Abstract

Elliptically contoured distributions generalize the multivariate normal distributions in such a way that the density generators need not be exponential. However, as the name suggests, elliptically contoured distributions remain to be restricted in that the proportional density level sets ought to be ellipsoids. In star-shaped distributions, this restriction is relaxed and the density level sets are allowed to be boundaries of arbitrary proportional star-shaped sets. In this note, we propose a non parametric estimator of the shape of density level sets of star-shaped distributions, and prove its strong consistency with respect to the Hausdorff distance. We illustrate our estimator with simulated and real data.

Keywords:

• Density level set
• direction
• elliptically contoured distribution
• Hausdorff distance
• kernel density estimator
• star-shaped distribution

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H12
• 62H11
• 62G07

Acknowledgements

The author is very grateful to the anonymous reviewer for his/her valuable comments and suggestions which led to much improvement on the earlier version of this paper.

Notes

1 C. Agostinelli and U. Lund (2013). R package circular: Circular Statistics (version 0.4-7). URL https://r-forge.r-project.org/projects/circular/.

2 2 Steve Kalke (2015). pgnorm: The p-Generalized Normal Distribution. R package version 2.0. https://CRAN.R-project.org/package=pgnorm.

Additional information

Funding

This work was supported by JSPS KAKENHI Grant Number 25400201.