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

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.