Falk, M.; Marohn, F. Laws of small numbers: Some applications to conditional curve estimation. (English) Zbl 0777.62038 Probability theory and applications, Essays to the Mem. of J. Mogyoródi, Math. Appl. 80, 257-278 (1992). [For the entire collection see Zbl 0755.00022.]Statistical inference from a sample \((X_ 1,Y_ 1,\dots,X_ n,Y_ n)\) for a functional parameter of the conditional distribution function \(F(.\mid x)\) of \(Y_ 1\) given \(X_ 1=x\) is obviously based on those \(Y_ i\) whose corresponding \(X_ i\)-values are close to \(x\). These data sets can be described by a truncated empirical point process.Using the Poisson approximation the authors replace this process by a Poisson process. The approximation error is characterized by bounds on the Hellinger distance between the distributions of the considered processes. The approximation entails that one can handle the relevant observations among the original sample like ideal observations whose stochastic behavior depends solely upon a few (unknown) parameters. This approach permits the application of standard methods to statistical questions concerning the original and typically nonparametric sample.In the present article it is demonstrated how this approach works in regression analysis; it is utilized to formulate results on the asymptotic normality of kernel estimates of general regression functionals and to derive asymptotically optimal estimators in the semiparametric setup. Reviewer: H.Liero (Potsdam) MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics Keywords:conditional curve estimation; semiparametric regression; local asymptotic normality; functional parameter; conditional distribution; truncated empirical point process; Poisson approximation; Poisson process; approximation error; Hellinger distance; asymptotic normality of kernel estimates of general regression functionals; asymptotically optimal estimators Biographic References: Mogyoródi, József Citations:Zbl 0755.00022 PDFBibTeX XMLCite \textit{M. Falk} and \textit{F. Marohn}, in: Probability theory and applications. Essays to the memory of József Mogyoródi. Dordrecht: Kluwer Academic Publishers. 257--278 (1992; Zbl 0777.62038)