Data-driven density estimation in the presence of additive noise with unknown distribution

We study the model Y = X + ε. We assume that we have at our disposal independent identically distributed observations Y1...,Yn and ε-1..,e-M. The (Xj)1≤j≤n are independent identically distributed with density fε, independent of the (εj)1≤j≤n,independent identically distributed with density fε. The a...

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Bibliographic Details
Published in:Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2011-09, Vol.73 (4), p.601-627
Main Authors: Comte, F., Lacour, C.
Format: Article
Language:English
Subjects:
Adaptive estimation
Additives
Computer simulation
Deconvolution
Density
Density estimation
Distribution
Distribution theory
Empirical analysis
Estimates
Estimating techniques
Estimation
Estimation methods
Estimators
Exact sciences and technology
Experimenters
Fourier transformations
General topics
Integers
Mathematical procedures
Mathematics
Mean-square risk
Measurement
Methodology
Minimax
Minimax rates
Noise
Non-parametric methods
Nonparametric inference
Observation
Parametric inference
Point estimators
Probability and statistics
Probability theory and stochastic processes
Projection
Sciences and techniques of general use
Signal noise
Simulation
Statistical methods
Statistics
Studies
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Summary:We study the model Y = X + ε. We assume that we have at our disposal independent identically distributed observations Y1...,Yn and ε-1..,e-M. The (Xj)1≤j≤n are independent identically distributed with density fε, independent of the (εj)1≤j≤n,independent identically distributed with density fε. The aim of the paper is to estimate without knowing fε. We first define an estimator, for which we provide bounds for the integrated L² -risk. We consider ordinary smooth and supersmooth noise ε with regard to ordinary smooth and supersmooth densities fε. Then we present an adaptive estimator of the density of fε. This estimator is obtained by penalization of a projection contrast and yields to model selection. Lastly, we present simulation experiments to illustrate the good performances of our estimator and study from the empirical point of view the importance of theoretical constraints.
ISSN:1369-7412
1467-9868
DOI:10.1111/j.1467-9868.2011.00775.x