Spline estimation of generalised monotonic regression

Abstract

We develop a simple and practical, yet flexible spline estimation method for semiparametric generalised linear models with monotonicity constraints. We propose to approximate the unknown monotone function by monotone B-splines, and employ generalised Rosen algorithm to compute the estimates. We show that the spline estimate of the nonparametric component achieves the optimal rate of convergence under the smooth condition, and that the estimates of regression parameters are asymptotically normal and efficient. The spline-based semiparametric likelihood ratio test (LRT) is also established. Moreover, a direct variance estimation method based on least-squares estimation is proposed. The finite sample performance of the spline estimates is evaluated by a Monte Carlo study. The methodology is illustrated on an air pollution study.

Keywords:

• air pollution
• generalised linear models
• generalised Rosen algorithm
• maximumlikelihood estimation
• monotone B-spline
• Monte Carlo

Acknowledgements

The author is grateful to the Editor, an Associate Editor, and two reviewers for their useful comments and constructive suggestions that led to significant improvement in the revised manuscript.