Estimation in monotone single‐index models

Single‐index models are popular regression models that are more flexible than linear models and still maintain more structure than purely nonparametric models. We consider the problem of estimating the regression parameters under a monotonicity constraint on the unknown link function. In contrast to the standard approach of using smoothing techniques, we review different “non‐smooth” estimators that avoid the difficult smoothing parameter selection. For about 30 years, one has had the conjecture that the profile least squares estimator is an n‐consistent estimator of the regression parameter, but the only non‐smooth argmin/argmax estimators that are actually known to achieve this n‐rate are not based on the nonparametric least squares estimator of the link function. However, solving a score equation corresponding to the least squares approach results in n‐consistent estimators. We illustrate the good behavior of the score approach via simulations. The connection with the binary choice and current status linear regression models is also discussed.