On deconvolution with repeated measurements

Abstract

In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of ``intrinsic'' values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same first-order properties as conventional estimators in the known-error case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothing-parameter choice.

Published in at http://dx.doi.org/10.1214/009053607000000884 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (22)

• A new multivariate measurement error model with zero-inflated dietary data, and its application to dietary assessment
• Density estimation with heteroscedastic error
• Möbius deconvolution on the hyperbolic plane with application to impedance density estimation
• Adaptive quantile estimation in deconvolution with unknown error distribution
• Deconvolution for an atomic distribution
• Minimax fast rates for discriminant analysis with errors in variables
• Laplace deconvolution and its application to Dynamic Contrast Enhanced imaging
• Distributionally-Robust Optimization with Noisy Data for Discrete Uncertainties Using Total Variation Distance
• A frequency domain analysis of the error distribution from noisy high-frequency data
• Joint self-supervised blind denoising and noise estimation
• Inference based on Kotlarski's Identity
• Uniform confidence bands in deconvolution with unknown error distribution
• Uniform confidence bands for nonparametric errors-in-variables regression
• Nonparametric modal regression in the presence of measurement error
• Deconvolution with Unknown Error Distribution Interpreted as Blind Isotonic Regression
• An alternative local polynomial estimator for the error-in-variables problem
• Conditional density estimation with covariate measurement error
• Deconvolution density estimation with penalised MLE
• Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging
• Phase Function Density Deconvolution with Heteroscedastic Measurement Error of Unknown Type
• Noisy classification with boundary assumptions
• Density deconvolution from repeated measurements without symmetry assumption on the errors