Rank test of hypothesis of randomness against a group of regression alternatives

summary:The problem of testing hypothesis of randomness against a group of alternatives of regression in a parameter is investigated and a rank test for this problem is suggested. This problem is a generalization of the problem of detecting a shift in a location parameter of a distribution occurring at an unknown time point between consecutively taken observations. The rank test in this work is shown to be locally average most powerful within the class of all possible rank tests in the sense of the definition in Section §3. The asymptotic normality of the rank test statistic and the asymptotic efficiency of the rank test are shown not only for the case of location and scale parameter but for the case of general parameter