Estimation Combining Unbiased and Possibly Biased Estimators

Abstract

Statisticians often face a dilemma, namely how to decide between choosing a parametric versus a nonparametric statistical model. Parametric statistical models can be asymptotically efficient if the model assumptions hold but biased under model misspecification. Nonparametric models, on the other hand, are often asymptotically unbiased but likely to be less efficient than parametric models if the parametric model is correctly specified. In this work, we propose a new estimator, which combines parametric and nonparametric estimators into a single estimating procedure. In contrast to previously suggested combined estimators, the new estimator ensures that it is asymptotically equivalent to the efficient parametric estimator if parametric assumptions hold. If the assumptions are violated, the combined estimator converges to the nonparametric estimator. The new combined estimator is illustrated with quantile regressions which estimate individualized prediction intervals. Through simulation studies, we explored the MSE of the proposed and competing estimators when data came from parametric distributions and, alternatively, when parametric assumptions were violated. The development of this approach was motivated by the real-data problem of predicting realistic hemoglobin A1C ranges in type 2 diabetes (T2DM) patients. A cross-validation confirmed that the proposed estimator shows better or similar prediction properties in comparison with several competing approaches.