This paper characterizes the bandwidth value (h) that is optimal for estimating parameters of the form $\eta \, = \,E\left[ {\omega /f_{V|U} \left({V|U} \right)} \right]$, where the conditional density of a scalar continuous random variable V, given a random vector U, $f_{V|U} $, is replaced by its kernel estimator. That is, the parameter η is the expectation of ω inversely weighted by $f_{V|U} $, and it is the building block of various semiparametric estimators already proposed in the literature such as Lewbel (1998), Lewbel (2000b), Honoré and Lewbel (2002), Khan and Lewbel (2007), and Lewbel (2007). The optimal bandwidth is derived by minimizing the leading terms of a second-order mean squared error expansion of an in-probability approximation of the resulting estimator with respect to h. The expansion also demonstrates that the bandwidth can be chosen on the basis of bias alone, and that a simple “plug-in” estimator for the optimal bandwidth can be constructed. Finally, the small sample performance of our proposed estimator of the optimal bandwidth is assessed by a Monte Carlo experiment.
Continuous Random Variable Estimation is not Optimal for the Witsenhausen Counterexample
