First and Second Order Asymptotic Bias Correction of Nonlinear Estimators in a Non-Parametric Setting and an Application to the Smoothed Maximum Score Estimator

This paper derives, extending the work of Rilstone, Srivastava and Ullah (1996), an analytical expression that takes account of first and second order asymptotic bias of nonlinear estimators in a non-parametric setting. By using moment expansions, we obtain a first and a second order bias removal mechanism. We specialize our results on the smoothed maximum score estimator of the coefficient vector of a binary response model in the dynamic setting of De Jong and Woutersen (2009). First order asymptotic theory has already been provided, although very large samples are needed to reach the asymptotic outcome of normality in this model. We provide a second order asymptotic expansion and, with the appropriate estimated quantities, we design a new bias-corrected estimator. Finally, a simulation study shows the advantages of our proposed bias-correction procedure.