Locally Most Powerful Tests Based on Sequential Ranks

Sequential ranks of data X1, X2, . . . observed sequentially in time are defined as ranks computed from the data observed so far, denoting Rii the rank of Xi among the values X1, X2, . . . , Xi for any i. This paper studies tests of various hypotheses based on sequential ranks and derives such tests, which are locally most powerful among all tests based on sequential ranks. Such locally most powerful sequential rank test is derived for the hypothesis of randomness against a general alternative, including the regression in location as a special case for the alternative hypothesis. Further, the locally most powerful sequential rank tests are derived for independence of two samples. The new tests are suitable for the situation when data are observed sequentially in time and the test is carried out each time after obtaining a new observation. While the classical rank tests require to recalculate all values of the ranks each time, the methods based on sequential ranks only require to compute the sequential rank of the only one new observation.