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Low order approximations in deconvolution and regression with errors in variables

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  • Raymond J. Carroll
  • Peter Hall

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.

Suggested Citation

  • Raymond J. Carroll & Peter Hall, 2004. "Low order approximations in deconvolution and regression with errors in variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 31-46.
  • Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:31-46
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9868.2004.00430.x
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    References listed on IDEAS

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    1. John Staudenmayer & David Ruppert, 2004. "Local polynomial regression and simulation-extrapolation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 17-30.
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    Cited by:

    1. Eric Weese & Masayoshi Hayashi & Masashi Nishikawa, 2015. "Inefficiency and Self-Determination: Simulation-based Evidence from Meiji Japan," Discussion Paper Series DP2015-35, Research Institute for Economics & Business Administration, Kobe University.
    2. William Horrace & Christopher Parmeter, 2011. "Semiparametric deconvolution with unknown error variance," Journal of Productivity Analysis, Springer, vol. 35(2), pages 129-141, April.
    3. Carrasco, Marine & Florens, Jean-Pierre, 2011. "A Spectral Method For Deconvolving A Density," Econometric Theory, Cambridge University Press, vol. 27(03), pages 546-581, June.
    4. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, number 8355, June.
    5. 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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