Estimating a Concave Distribution Function from Data Corrupted with Additive Noise

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...

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Bibliographic Details
Published in:The Annals of statistics 2009-04, Vol.37 (2), p.782-815
Main Authors: Jongbloed, Geurt, van der Meulen, Frank H.
Format: Article
Language:English
Subjects:
62E20
62G05
Algorithms
Asymptotic distribution
Asymptotic methods
Decision theory
deconvolution
decreasing density
Density estimation
Distribution functions
Distribution theory
Estimation methods
Estimators
Exact sciences and technology
General topics
Least squares
Mathematics
maximum likelihood
Maximum likelihood estimation
Maximum likelihood estimators
Maximum likelihood method
Minimax
minimax risk
Objective functions
Parameter estimation
Probability and statistics
Probability theory and stochastic processes
Random variables
Sciences and techniques of general use
Statistics
Studies
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/07-AOS579