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J R Statist Soc B 2004 66 Part 1 pp 31 46 Low order approximations in deconvolution and regression with errors in variables Raymond J Carroll Texas A M University College Station USA and Peter Hall Australian National University Canberra Australia Received September 2002 Revised April 2003 Summary 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 Keywords Density estimation Measurement error Nonparametric regression Orthogonal series Simulation extrapolation 1 Introduction Suppose that we observe the value of W X U 1 where the random variables X and U are independently distributed We either know or have data on the distribution of U and we

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