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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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