Abstract
SIMEX is a general-purpose technique for measurement error correction. There is a substantial literature on the application and theory of SIMEX for purely parametric problems, as well as for purely non-parametric regression problems, but there is neither application nor theory for semiparametric problems. Motivated by an example involving radiation dosimetry, we develop the basic theory for SIMEX in semiparametric problems using kernel-based estimation methods. This includes situations that the mismeasured variable is modeled purely parametrically, purely non-parametrically, or that the mismeasured variable has components that are modeled both parametrically and nonparametrically. Using our asymptotic expansions, easily computed standard error formulae are derived, as are the bias properties of the nonparametric estimator. The standard error method represents a new method for estimating variability of nonparametric estimators in semiparametric problems, and we show in both simulations and in our example that it improves dramatically on first order methods.We find that for estimating the parametric part of the model, standard bandwidth choices of order O(n(-1/5)) are sufficient to ensure asymptotic normality, and undersmoothing is not required. SIMEX has the property that it fits misspecified models, namely ones that ignore the measurement error. Our work thus also more generally describes the behavior of kernel-based methods in misspecified semiparametric problems.
Highlights
Regression models with measurement errors arise frequently in practice and have attracted attention in the statistics literature
A semiparametric regression model with errors in variable has been considered by several authors in the attempt to develop a measurement error calibration when the errors are in the linear part of linear regression [17], or generalized linear regression [18]. [43] used a method of moments and deconvolution to construct the calibration for the case of partially linear models when the mismeasured covariate appears in parametric and nonparametric parts
We show using our example that the standard error method improves dramatically on first order methods for estimating standard errors of the nonparametric components, and is of importance even when there is no measurement error
Summary
In either the mixture of Berkson and classical errors, or when there is no Berkson error, and using either the excess relative risk model (15), a naive semiparametric fit that ignores the measurement error, or using the SIMEX semiparametric fit, the estimate of β1 ≈ 1.75 with a standard error (s.e.) ≈ 0.25. The left two panels give results for the mixture of Berkson errors model using parametric (top) and semiparametric (bottom) fits. The excess relative risk parameter estimates for the naive and measurement error analysis are, respectively, 6.24 and 7.75 for the mixture error model, and 7.83 and 12.60 for the no Berkson error model. The solid line is the usual estimated pointwise standard errors based upon the formal theory in Result 1 and equation (9). The bootstrap with 1000 bootstrap samples requires 401, 000 semiparametric fits
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