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.
"Estimating a concave distribution function from data corrupted with additive noise." Ann. Statist. 37 (2) 782 - 815, April 2009. https://doi.org/10.1214/07-AOS579