A new algorithm to estimate monotone nonparametric link functions and a comparison with parametric approach

The generalized linear model (GLM) is a class of regression models where the means of the response variables and the linear predictors are joined through a link function. Standard GLM assumes the link function is fixed, and one can form more flexible GLM by either estimating the flexible link function from a parametric family of link functions or estimating it nonparametically. In this paper, we propose a new algorithm that uses P-spline for nonparametrically estimating the link function which is guaranteed to be monotone. It is equivalent to fit the generalized single index model with monotonicity constraint. We also conduct extensive simulation studies to compare our nonparametric approach for estimating link function with various parametric approaches, including traditional logit, probit and robit link functions, and two recently developed link functions, the generalized extreme value link and the symmetric power logit link. The simulation study shows that the link function estimated nonparametrically by our proposed algorithm performs well under a wide range of different true link functions and outperforms parametric approaches when they are misspecified. A real data example is used to illustrate the results.