Rank Score Comparison of Several Regression Parameters

Abstract

For testing $\beta_i = \beta, i = 1,\cdots, k$, in the model $Y_{ij} = \alpha + \beta_iX_{ij} + Z_{ij} j = 1,\cdots, n_i$ a class of rank score tests is presented. The test statistic is based on the simultaneous ranking of all the observations in the different $k$ samples. Its asymptotic distribution is proved to be chi square under the hypothesis and noncentral chi square under an appropriate sequence of alternatives. The asymptotic efficiency of the given procedure relative to the least squares procedure is shown to be the same as the efficiency of rank score tests relative to the $t$-test in the two sample problem. The proposed criterion would be an asymptotically most powerful rank score test for the hypothesis if the distribution function of the observations is known.