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April 2015 On nonnegative unbiased estimators
Pierre E. Jacob, Alexandre H. Thiery
Ann. Statist. 43(2): 769-784 (April 2015). DOI: 10.1214/15-AOS1311


We study the existence of algorithms generating almost surely nonnegative unbiased estimators. We show that given a nonconstant real-valued function $f$ and a sequence of unbiased estimators of $\lambda\in\mathbb{R}$, there is no algorithm yielding almost surely nonnegative unbiased estimators of $f(\lambda)\in\mathbb{R}^{+}$. The study is motivated by pseudo-marginal Monte Carlo algorithms that rely on such nonnegative unbiased estimators. These methods allow “exact inference” in intractable models, in the sense that integrals with respect to a target distribution can be estimated without any systematic error, even though the associated probability density function cannot be evaluated pointwise. We discuss the consequences of our results on the applicability of pseudo-marginal algorithms and thus on the possibility of exact inference in intractable models. We illustrate our study with particular choices of functions $f$ corresponding to known challenges in statistics, such as exact simulation of diffusions, inference in large datasets and doubly intractable distributions.


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Pierre E. Jacob. Alexandre H. Thiery. "On nonnegative unbiased estimators." Ann. Statist. 43 (2) 769 - 784, April 2015.


Published: April 2015
First available in Project Euclid: 3 March 2015

zbMATH: 1321.65015
MathSciNet: MR3319143
Digital Object Identifier: 10.1214/15-AOS1311

Primary: 65C50 , 65C60 , 68W20

Keywords: Bernoulli factory , Monte Carlo methods , Poisson estimator , sign problem , unbiased estimator

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 2 • April 2015
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