The Annals of Statistics

Estimating a concave distribution function from data corrupted with additive noise

Geurt Jongbloed and Frank H. van der Meulen

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We consider two nonparametric procedures for estimating a concave distribution function based on data corrupted with additive noise generated by a bounded decreasing density on (0, ∞). For the maximum likelihood (ML) estimator and least squares (LS) estimator, we state qualitative properties, prove consistency and propose a computational algorithm. For the LS estimator and its derivative, we also derive the pointwise asymptotic distribution. Moreover, the rate n−2/5 achieved by the LS estimator is shown to be minimax for estimating the distribution function at a fixed point.

Article information

Ann. Statist., Volume 37, Number 2 (2009), 782-815.

First available in Project Euclid: 10 March 2009

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Zentralblatt MATH identifier

Primary: 62E20: Asymptotic distribution theory 62G05: Estimation

Asymptotic distribution deconvolution decreasing density least squares maximum likelihood minimax risk


Jongbloed, Geurt; van der Meulen, Frank H. Estimating a concave distribution function from data corrupted with additive noise. Ann. Statist. 37 (2009), no. 2, 782--815. doi:10.1214/07-AOS579.

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