Estimation of Star-Shaped Distributions

Abstract

Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets.

Modelling with star-shaped distributions

“…For understanding normalizing constants of density generating functions as ball numbers (even in more general cases), we refer to [21]. Semiparametric and parametric estimation methods for density generators, generalized radius distributions and the star-shaped densities are examined in the paper [11]. The latter three papers contain references to a lot of other papers on star-shaped distributions and ball numbers.…”

High-dimensional nonparametric density estimation via symmetry and shape constraints

“…In the former case, for uniformly bounded densities, Biau and Devroye (2003) established a minimax lower bound in total variation distance of order n −1/(p+2) , while in the latter case, the main interest has been in the classes with s < 0, which contain the class F p , so the same minimax lower bounds apply as for F p . Various other simplifying structures and methods have also been considered for nonparametric high-dimensional density estimation, including kernel approaches for forest density estimation (Liu et al, 2011) and star-shaped density estimation (Liebscher and Richter, 2016), as well as nonparametric maximum likelihood methods for independent component analysis (Samworth and Yuan, 2012). Perhaps most closely related to this work is the approach of Bhattacharya and Bickel (2012), who consider a maximum likelihood approach (as well as spline approximations) to estimating the generator of an elliptically symmetric distribution with decreasing generator.…”

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

“…In a recent paper, Liebscher and Richter [8] presented examples of parametric modeling and estimation concerning the shape of the density contours of two-dimensional star-shaped distributions (Section 2.2 as well as Sections 3.3 and 3.4 of [8]). They also investigated estimation about many other aspects of star-shaped distributions.…”