# Estimation of the shape of the density contours of star-shaped distributions

## TL;DR

This paper introduces a nonparametric estimator for the shape of density contours in star-shaped distributions, extending beyond elliptical contours, and proves its strong consistency with simulation illustrations.

## Contribution

It proposes a novel nonparametric estimator for the shape of density contours in star-shaped distributions, with proven strong consistency.

## Key findings

- Estimator is strongly consistent under Hausdorff distance
- Simulation demonstrates estimator's practical effectiveness
- Extends contour shape estimation beyond elliptical 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 similar density contours ought to be elliptical. Kamiya, Takemura and Kuriki [Star-shaped distributions and their generalizations, Journal of Statistical Planning and Inference 138 (2008), 3429--3447] proposed star-shaped distributions, for which the density contours are allowed to be boundaries of arbitrary similar star-shaped sets. In the present paper, we propose a nonparametric estimator of the shape of the density contours of star-shaped distributions, and prove its strong consistency with respect to the Hausdorff distance. We illustrate our estimator by simulation.