Title: Estimation of the shape of the density contours 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 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.