Galton's test as a linear rank test with estimated scores and its local asymptotic efficiency
On the basis of two independent samples \(X_ 1,...,X_ m\) (from F) and \(Y_ 1,...,Y_ n\) (from G) \((n+m=N)\) the hypothesis \(H_ 0:F=G\) has to be tested against the nonparametric alternative \(H_ 1:F\leq G\), \(F\neq G\). In this paper the authors propose a linear rank test with estimated scores, which is obtained from the rank statistic \[ S_ b = \frac{n}{N} \sum^{m}_{i=1} (\bar f-\bar g) \left( \frac{R_{1i}}{N+1}\right) - \frac{m}{N} \sum^{n}_{j=1} (\bar f-\bar g) \left(\frac{R_{2j}}{N+1}\right) \] by estimating the score function \(b=\bar f-\bar g\). Here \(\bar f = d(F\circ H^{-1})/dx\), \(\bar g = d(G\circ H^{-1})/dx\), \(H = (mF+nG)/(m+n)\) and the \(R_{1i}\)'s and the \(R_{2j}\)'s denote the ranks of the \(X_ i\)'s and the \(Y_ j\)'s in the combined sample. Replacing now b by a ''natural'' estimator \(\hat b\) based upon the ''empirical two sample rank process'', the authors obtain a linear rank test with estimated scores, which turns out to be (closely related to) the well-known Galton test. By a study of the local asymptotic efficiency [in the sense of \textit{J. Hájek} and \textit{Z. Šidák}, Theory of rank tests. (1967; Zbl 0161.381)] for a number of local alternatives it is revealed that the power properties of the authors' test are not very good. This is confirmed by a Monte Carlo study of the power for some fixed alternatives. It is concluded that the proposed test does not yield an adaptive procedure; apparently the estimation of the score function \(b\) is too crude.
scientific article; zbMATH DE number 3842990