Abstract
We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.
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Acknowledgments
Thanks to two referees, whose comments lead to various improvements in the text and streamlining of the proof of Theorem 2.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Jongbloed, G. Consistent likelihood-based estimation of a star-shaped distribution. Metrika 69, 265–282 (2009). https://doi.org/10.1007/s00184-008-0217-0
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DOI: https://doi.org/10.1007/s00184-008-0217-0