Dimension reduction in semiparametric measurement error models with errors in covariates

Vita.In many regression models one or more of the covariates are measured with error. It is well known that in such situations, traditional estimation procedures using the mismeasured covariates can lead to misleading results. In particular we consider the semiparametric logistic regression model in which a binary response Y is related to a scalar predictor X and a vector of additional predictors Z, where X is measured with error and a surrogate W. We will make no parametric assumptions on the distribution of X given (W, Z). Existing techniques rely on a kernel regression of the "true" covariate on all the observed covariates and surrogates. This requires a nonparametric regression in as many dimensions as there are covariates and surrogates. The usual theory copes with such higher dimensional problems by using higher order kernels, but this seems unrealistic for most problems. We will show that the usual theory is essentially as good as one can do with this technique. Instead of regression with higher order kernels, we propose the use of dimension reduction techniques. We first suppose that the "true" covariate depends only on a linear combination of the observed covariates and surrogates. If this linear combination were known, we could apply the one-dimensional versions of the semiparametric problem, for which standard kernels are applicable. We show that if one can estimate the linear directions at the root-n rate, then asymptotically the resulting estimator behaves as if the linear combination were known. We will also show how these dimension reduction techniques can be applied to some existing semiparametric methods to estimate the slope parameters and their covariance. Finally, we present a simulation study to evaluate the estimators under some models