Principal Hessian Directions for regression with measurement error

We consider a nonlinear regression problem with predictors with measurement error. We assume that the response is related to unknown linear combinations of a p-dimensional predictor vector through an unknown link function. Instead of observing the predictors, we observe a surrogate vector with the property that its expectation is linearly related to the predictor vector with constant variance. We use an important linear transformation of the surrogates. Based on the transformed variables, we develop the modified Principal Hessian Directions method for estimating the subspace of the effective dimension-reduction space. We derive the asymptotic variances of the modified Principal Hessian Directions estimators. Several examples are reported and comparisons are made with the sliced inverse regression method of Carroll & Li (1992). Copyright Biometrika Trust 2004, Oxford University Press.