Аннотация:
Semiparametric Models are characterized by an infinite dimensional parameter, while the target of estimation is only a finite - often low - dimensional. A prominent example is the estimation of a finite dimensional projection of the full parameter via an M-Estimator, as for example the profile Maximum Likelihood Estimator (pMLE). Despite the full model being nonparametric root n rates can be attained for such estimators. The semiparametric Wilks and Fisher Theorems show that the semiparametric log likelihood quotient is asymptotically chi square distributed - the degrees of freedom equal the dimension of the target parameter - and that the pMLE is semiparametrically efficient. We present a method how to extend these results to a non asymptotic setting and how to obtain explicit bounds for the "small terms". This allows to determine for a broad class of models critical ratios of the full dimension to the sample size in the context of sieve estimators. The results are illustrated with the single index model.
