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An event encountered for the first time can become a common phenomenon in a population through transmission over time. This study focuses on nonparametric partially sequential test statistics in order to early detect such events. For this purpose, firstly, the partially sequential test statistic given in Mukherjee [1] is considered. Then, with a modification of this statistic, a new nonparametric partially sequential test statistic that works at almost the desired alpha level of significance is revealed. Also, based on Monte-Carlo simulations, the proposed test statistic is shown to be more powerful than Mukherjee's statistic. Lastly, two illustrative numerical examples are also given.

We propose a sequential procedure for detecting a possible changepoint in a random sequence of observations so that we can fix the probabilty of stopping at any level if there is no change, while otherwise we will stop with probabilty one in a specified length of time.

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist. 9 424–436) and Lombard (1981, South African Statist. J. 15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.

A technique is introduced to prove limit theorems for simple linear rank statistics by means of an approximating statistic based on sequential ranks. This approximation is shown to be close enough to prove asymptotic normality of simple linear rank statistics under the null hypothesis and to obtain a bound on the rate of convergence to normality when the score function is unbounded. In addition, a law of the iterated logarithm and an invariance principle are given for simple linear rank statistics.