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# maca.biometrika

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NONPARAMETRIC REGRESSION IN THE PRESENCE OF MEASUREMENT ERROR BY RAYMOND J CARROLL Department of Statistics Texas A M University College Station TX 77843 3143 U S A carroll stat tamu edu JEFFREY D MACA Department of Biostatistics Novartis Pharmaceuticals Corporation 59 Route 10 East Hanover NJ 07936 1080 U S A jeff maca pharma novartis com AND DAVID RUPPERT School of Operations Research and Industrial Engineering Cornell University Ithaca NY 14853 U S A davidr orie cornell edu SUMMARY In many regression applications the independent variable is measured with error When this happens conventional parametric and nonparametric regression techniques are no longer valid We consider two different approaches to nonparametric regression The first uses the SIMEX method and makes no assumption about the distribution of the unobserved error prone predictor For this approach we derive an asymptotic theory for kernel regression which has some surprising implications Penalised regression splines are also considered for fixed number of known knots The second approach assumes that the error prone predictor has a distribution of a mixture of normals with an unknown number of mixtures and uses regression splines Simulations illustrate the results Some key words Estimating equation Local polynomial regression Measurement error Regression spline Sandwich estimation SIMEX Short title Nonparametric Regression with Measurement Error 1 INTRODUCTION We consider the problem of nonparametric regression function estimation in the presence of measurement error in the predictor Suppose that the regression of a response Y on a predictor X is given by E Y X m X Instead of observing X we can only observe W an error prone measurement related to X by an additive error model W X U where U is a mean zero normal random variable with variance u2 The question is how to estimate m when observations on Y and W are all that are available This problem has been addressed previously most notably by Fan Truong

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