On deconvolution with repeated measurements

By Aurore Delaigle, Peter Hall, A. Meister. 2008

215 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2f450ea90bc80b4d7c53040a9c4a778dd081b18f

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Citations in the corpus

Methodology for non‐parametric deconvolution when the error distribution is unknown (2016, methodology): “Li and Vuong (1998), Lin and Carroll (2006), Hall and Ma (2007), Delaigle et al. (2008) and Stefanski and McIntyre (2011) addressed measurement error problems in the setting of replicated data, and Diggle and Hall (1993) and Neumann (1997) considered cases where samples of error data are available.”
Nonparametric Regression Analysis for Group Testing Data (2011, background): “(See Delaigle, Hall, and Meister 2008 for details and conditions.)”
Nonparametric Regression Estimation in the Heteroscedastic Errors-in-Variables Problem (2007, methodology): “Within each group, we use the technique developed by Delaigle et al. (2007a), that is, we estimate f g,ftU (t) by ̂f g,ftU (t) = |1/Ng ∑ (j,k1,k2)∈Sg cos{t (Wj,k1 − Wj,k2)}|1/2, where Sg = {(j, k1, k2) s.t. 1 ≤ j ≤ n, 1 ≤ k1 < k2 ≤ rj and Ij = g} and Ng = #Sg .”
Nonlinear Panel Data Estimation via Quantile Regressions (2015, background): “(23) The goal is the identification of fY1|η,X , fY2|η,X , fY3|η,X and fη|X given knowledge of fY1,Y2,Y3|X .”
Goodness-of-fit tests in mixed models (2009, methodology): “This fact has been exploited in recent work by Li and Vuong (1998), Hall and Yao (2003), Delaigle et al. (2008) and Hart and Cañette (2008), all of whom propose methods for estimating f and g in model (9).”
Identifying distributional characteristics in random coefficients panel data models (2009, background): “…the distribution of errors from within-group equations in a simple model with an individual-specific intercept and symmetric errors, see also Li and Vuong (1998), Hall and Yao (2003), Delaigle et al. (2008), and Bonhomme and Robin (2009b) for related approaches in similar or more general models.”
A Few Bad Apples? Racial Bias in Policing (2017, methodology): “20This approach is a discretized version of a deconvolution procedure (Delaigle et al., 2008).”