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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.

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