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Estimation of semiparametric models when the criterion function is not smooth. (English) Zbl 1154.62325
Summary: We provide easy to verify sufficient conditions for the consistency and asymptotic normality of a class of semiparametric optimization estimators where the criterion function does not obey standard smoothness conditions and simultaneously depends on some nonparametric estimators that can themselves depend on the parameters to be estimated. Our results extend existing theories such as those of {\it A. Pakes} and {\it D. Pollard} [Econometrica 57, No. 5, 1027--1057 (1989; Zbl 0698.62031)], {\it D. W. K. Andrews} [Econometrica 62, No. 1, 43--72 (1994; Zbl 0798.62104)], and {\it W. K. Newey} [Econometrica 62, No. 6, 1349--1382 (1994; Zbl 0816.62034)]. We also show that bootstrap provides asymptotically correct confidence regions for the finite dimensional parameters. We apply our results to two examples: a `hit rate’ and a partially linear median regression with some endogenous regressors.

##### MSC:
 62G05 Nonparametric estimation 62G08 Nonparametric regression
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