On deconvolution with repeated measurements

TL;DR

This paper demonstrates that in statistical inverse problems, repeated measurements enable consistent estimation of error distributions, allowing kernel estimators to perform as well as if the error distribution were known, even with few replications.

Contribution

It introduces methods for nonparametric deconvolution using replicated data, achieving performance comparable to known-error scenarios with minimal replications.

Findings

Kernel estimators can match known-error performance with replicated data.

Estimators can be constructed with properties similar to conventional ones without knowing the error distribution.

Practical smoothing-parameter rules are proposed for these estimators.

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.