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Asymptotically optimal rank tests for the two-sample problem with randomly censored data. (English) Zbl 0658.62063
The paper is devoted to the asymptotic study of the two-sample testing problem under random censorship of the data. For m,n$$\geq 1$$, let $$X_{j1},...,X_{jm},X_{jm+1},...,X_{jN}$$, $$j=1,2$$, be independent r.v.’s. For $$j=1$$ the $$X_{ji}'s$$ are distributed according to a continuous d.f. $$F(x-c_{Ni}\Theta)$$ for some unknown $$\Theta \in R^ 1$$, $$c_{Ni}$$ being the scores. For $$j=2$$ the r.v.’s $$X_{ji}'s$$ are not observable, unlike the r.v.’s $$X_ i=\min (X_{1i},X_{2i})$$ and the censoring r.v.’s $$\Delta_ i=I(X_{1i}\leq X_{2i})$$, the indicator functions of the events $$\{X_{1i}\leq X_{2i}\}$$, $$1\leq i\leq N$$. The d.f. H of $$X_ i$$ for $$\Theta =0$$ is $$H=1-(1-F)(1-F_ 2),$$ and, for $$\Theta =0$$, $$\Delta_ i$$ has binomial (1,$$\pi)$$ distribution with $\pi =\int (1-F_ 2)dF=\int F dF_ 2.$ Asymptotically optimal rank tests for testing the null-hypothesis $$H_ 0:\Theta =0$$ versus the alternative $$H_ 1:\Theta >0$$ on the basis of the observable observations $$Z_ i=(X_ i,\Delta_ i)$$, $$1\leq i\leq N$$, are constructed and their properties studied. The proposed tests are distribution free under $$H_ 0$$, i.e., independent of F and of the censoring d.f. $$F_ 2$$, and are asymptotically optimal for certain fixed $$F,F_ 2$$. The form of the test statistics $$S_ N$$ is as simple as in the uncensored case, namely $S_ N=\sum^{n}_{i=1}C_{Ni}\{a^{(1)}_{NR_ i}\Delta_ i+a^{(2)}_{NR_ i}(1-\Delta_ i)\},$ where $$a_{Ni}^{(j)}$$, $$1\leq i\leq N$$, $$j=1,2$$, are two sets of scores.
Reviewer: J.Antoch

##### MSC:
 62G10 Nonparametric hypothesis testing
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##### References:
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