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May 2021 Bayesian estimation of a decreasing density
Geurt Jongbloed, Frank van der Meulen, Lixue Pang
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Braz. J. Probab. Stat. 35(2): 392-420 (May 2021). DOI: 10.1214/20-BJPS482


Suppose X1,,Xn is a random sample from a bounded and decreasing density f0 on [0,). We are interested in estimating such f0, with special interest in f0(0). This problem is encountered in various statistical applications and has gained quite some attention in the statistical literature. It is well known that the maximum likelihood estimator is inconsistent at zero. This has led several authors to propose alternative estimators which are consistent. As any decreasing density can be represented as a scale mixture of uniform densities, a Bayesian estimator is obtained by endowing the mixture distribution with the Dirichlet process prior. Assuming this prior, we derive contraction rates of the posterior density at zero by carefully revising arguments presented in Salomond (Electronic Journal of Statistics 8 (2014) 1380–1404). Several choices of base measure are numerically evaluated and compared. In a simulation various frequentist methods and a Bayesian estimator are compared. Finally, the Bayesian procedure is applied to current durations data described in Slama et al. (Human Reproduction 27 (2012) 1489–1498).


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Geurt Jongbloed. Frank van der Meulen. Lixue Pang. "Bayesian estimation of a decreasing density." Braz. J. Probab. Stat. 35 (2) 392 - 420, May 2021.


Received: 1 September 2019; Accepted: 1 July 2020; Published: May 2021
First available in Project Euclid: 24 March 2021

Digital Object Identifier: 10.1214/20-BJPS482

Keywords: Bayesian nonparametrics , contraction rate , posterior consistency

Rights: Copyright © 2021 Brazilian Statistical Association


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Vol.35 • No. 2 • May 2021
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