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)(Z,X), where Z is a real-valued co-variate with some unknown distribution, and the response X conditional on Z   is distributed according to the density p(·,ψ(Z))p(·,ψ(Z)), where p(·,θ)p(·,θ) is a one-parameter exponential family. The function ψψ is a smooth monotone function. Under this formulation, the regression function E(X|Z)E(X|Z) is monotone in the co-variate Z   (and can be expressed as a one–one function of ψψ); 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 ψψ 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 ψψ and also the regression function. The methodology is extended to the general heteroscedastic model, and its asymptotic properties are discussed.