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Two sample rank estimators of optimal nonparametric score-functions and corresponding adaptive rank statistics. (English) Zbl 0548.62029
Let \(X_ 1,...,X_ n\) and \(X_{n+1},...,X_{n+m}\) be two samples from d.f. F and G, respectively, and let \(Q_ i=R_ i/(n+m)\) be a normalized rank of \(X_ i\) among all X’s. Put \(H=\lambda F+(1-\lambda)G\) where \(n/(n+m)\to\lambda \) as \(n+m\to\infty \). The authors introduce interesting kernel estimations \(\hat f\) and \(\hat g\) based on ranks \(Q_ 1,...,Q_ n\) and \(Q_{n+1},...,Q_{n+m}\) for the densities \(f(t)=dF[H^{-1}(t)]/dt\) and \(g(t)=dG[H^{-1}(t)]/dt.\)
Consistency properties of \(\hat f\) and \(\hat g\) are studied and central limit theorems are proved for statistics \(\sum^{n}_{1}[\hat f(Q_ i)-g(Q_ i)]\) as a result of central limit theorems for more general statistics \(\sum^{n}_{1}{\hat\psi }(Q_ i)\) with \({\hat\psi }\) being estimators of the score functions \(\psi\). The theorems are proved for fixed F and G.
Reviewer: E.Khmaladze

62G05 Nonparametric estimation
62G99 Nonparametric inference
60F05 Central limit and other weak theorems
62G10 Nonparametric hypothesis testing
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