Estimation of smooth regression functions in monotone response models

We consider the estimation of smooth regression functions in a class of conditionally parametric co-variate-response models. Independent and identically distributed observations are available from the distribution of (Z, X), where Z is a real-valued covariate with some unknown distribution, and the response X conditional on Z is distributed according to the density p(., psi(Z)), where p(., 0) is a one-parameter exponential family. The function psi is a smooth monotone function. Under this formulation, the regression function E(X vertical bar Z) is monotone in the co-variate Z (and can be expressed as a one-one function of psi); hence the term "monotone response model". Using a penalized least squares approach that incorporates both monotonicity and smoothness, we develop a scheme for producing smooth monotone estimates of the regression function and also the function psi across this entire class of models. Point-wise asymptotic normality of this estimator is established, with the rate of convergence depending on the smoothing parameter. This enables construction of Wald-type (point-wise) as well as pivotal confidence sets for psi and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed. (C) 2008 Elsevier B.V. All rights reserved.