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Optimal Rates of Convergence for Deconvolving a Density

Raymond J. Carroll and Peter Hall
Journal of the American Statistical Association
Vol. 83, No. 404 (Dec., 1988), pp. 1184-1186
DOI: 10.2307/2290153
Stable URL: http://www.jstor.org/stable/2290153
Page Count: 3
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Optimal Rates of Convergence for Deconvolving a Density
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Abstract

Suppose that the sum of two independent random variables X and Z is observed, where Z denotes measurement error and has a known distribution, and where the unknown density f of X is to be estimated. One application is the estimation of a prior density for a sequence of location parameters. A second application arises in the errors-in-variables problem for nonlinear and generalized linear models, when one attempts to model the distribution of the true but unobservable covariates. This article shows that if Z is normally distributed and f has k bounded derivatives, then the fastest attainable convergence rate of any nonparametric estimator of f is only (log n)-k/2. Therefore, deconvolution with normal errors may not be a practical proposition. Other error distributions are also treated. Stefanski-Carroll (1987a) estimators achieve the optimal rates. The results given have versions for multiplicative errors, where they imply that even optimal rates are exceptionally slow.

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