Estimating a concave distribution function from data corrupted with additive noise

TL;DR

This paper introduces two nonparametric methods for estimating a concave distribution function from noisy data, establishing their properties, consistency, and optimal convergence rates.

Contribution

It presents new nonparametric estimators (ML and LS) for concave distribution functions under additive noise, with proven consistency and asymptotic properties.

Findings

Both estimators are consistent and have well-defined asymptotic distributions.

The LS estimator achieves the minimax rate of convergence $n^{-2/5}$ at a fixed point.

Computational algorithms for the estimators are proposed.

Abstract

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 $(0,\infty)$. 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.