SEMIPARAMETRIC QUASILIKELIHOOD AND VARIANCE FUNCTION ESTIMATION IN MEASUREMENT ERROR MODELS

We consider a quasilikelihood/variance function model when a predictor X is measured with error and a surrogate W is observed. When in addition to a primary data set containing (Y, W) a validation data set exists for which (X, W) is observed, we can (i) estimate the first and second moments of the response Y given W by kernel regression; (ii) use quasilikelihood and variance function techniques to estimate the regression parameters as well as variance structure parameters. The estimators are shown to be asymptotically normally distributed, with asymptotic variance depending on the size of the validation data set and not on the bandwith used in the kernel estimates. A more refined analysis of the asymptotic covariance shows that the optimal bandwidth converges to zero at the rate n-1/3.