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Low order approximations in deconvolution and regression with errors in variables

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  • Raymond J. Carroll
  • Peter Hall
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    Abstract

    We suggest two new methods, which are applicable to both deconvolution and regression with errors in explanatory variables, for nonparametric inference. The two approaches involve kernel or orthogonal series methods. They are based on defining a low order approximation to the problem at hand, and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently. Of course, both techniques could be employed to construct consistent estimators, but in many contexts of importance (e.g. those where the errors are Gaussian) consistency is, from a practical viewpoint, an unattainable goal. We rephrase the problem in a form where an explicit, interpretable, low order approximation is available. The information that we require about the error distribution (the error-in-variables distribution, in the case of regression) is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors, e.g. through repeated measurements. In particular, we do not need to estimate a function, such as a characteristic function, which expresses detailed properties of the error distribution. This feature of our methods, coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages, means that the methods are particularly economical in computing time. Copyright 2004 Royal Statistical Society.

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    Bibliographic Info

    Article provided by Royal Statistical Society in its journal Journal of the Royal Statistical Society Series B.

    Volume (Year): 66 (2004)
    Issue (Month): 1 ()
    Pages: 31-46

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    Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:31-46

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    Cited by:
    1. William C. Horrace & Christopher F. Parmeter, 2008. "Semiparametric Deconvolution with Unknown Error Variance," Center for Policy Research Working Papers 104, Center for Policy Research, Maxwell School, Syracuse University.
    2. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    3. Thomas, Laine & Stefanski, Leonard A. & Davidian, Marie, 2013. "Moment adjusted imputation for multivariate measurement error data with applications to logistic regression," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 15-24.

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