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Published online by Cambridge University Press:  18 August 2022

Chan Shen*
Penn State University
Roger Klein
Rutgers University
Address correspondence to Chan Shen, Departments of Surgery and Public Health Sciences, Penn State University, College of Medicine, Hershey, PA, USA; e-mail:
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Controlling the bias is central to estimating semiparametric models. Many methods have been developed to control bias in estimating conditional expectations while maintaining a desirable variance order. However, these methods typically do not perform well at moderate sample sizes. Moreover, and perhaps related to their performance, nonoptimal windows are selected with undersmoothing needed to ensure the appropriate bias order. In this paper, we propose a recursive differencing estimator for conditional expectations. When this method is combined with a bias control targeting the derivative of the semiparametric expectation, we are able to obtain asymptotic normality under optimal windows. As suggested by the structure of the recursion, in a wide variety of triple index designs, the proposed bias control performs much better at moderate sample sizes than regular or higher-order kernels and local polynomials.

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This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
© The Author(s), 2022. Published by Cambridge University Press


We thank the seminar participants at Columbia University and New York University for helpful comments and suggestions. We also thank the Editor and referees for their insightful comments and suggestions. The authors are solely responsible for any errors.



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