Estimation of Smooth Link Functions in Monotone Response Models

We deal with the estimation of smooth link functions in a class of conditionally parametric models, of the covariate-response type. 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(?; ?(Z)), where p(?; 碌) is a one-parameter exponential family, naturally parametrized. The function ? is a smooth monotone function. Under this formulation, the regression function E(X j Z) is monotone in the covariate Z (and can be expressed as a one-one function of ?); hence the term "monotone response model". Using a penalized least squares approach that incorporates both monotonicity and smoothness constraints, we develop a scheme for producing smooth monotone estimates of the function ? across this entire class of models. Pointwise asymptotic normality of this estimator is established, with the rate of convergence depending