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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

Biographic References:

Mogyoródi, József

Citations:

Zbl 0755.00022
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