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Risks 2016, 4(4), 44; doi:10.3390/risks4040044

Estimation of Star-Shaped Distributions

1
Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, 06217 Merseburg, Germany
2
Institute of Mathematics, University of Rostock, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany
*
Author to whom correspondence should be addressed.
Received: 2 September 2016 / Accepted: 18 November 2016 / Published: 30 November 2016
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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. View Full-Text
Keywords: star-shaped distributions; antinorm contoured distributions; norm contoured distributions; non-concentric elliptically contoured distributions; kernel density estimators star-shaped distributions; antinorm contoured distributions; norm contoured distributions; non-concentric elliptically contoured distributions; kernel density estimators
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Liebscher, E.; Richter, W.-D. Estimation of Star-Shaped Distributions. Risks 2016, 4, 44.

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