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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.

Distributions of exceedance statistics based on minimal spacing of generalized order statistics are obtained in a random threshold model. As the special cases of the generalized order statistics the ordinary order statistics, Progressively Type II right censored order statistics and record values are considered. The results obtained in the paper imply previous results on exceedance statistics for the variety of models of ordered random variables.

Distributions of exceedance statistics based on generalized order statistics are obtained for a random threshold model. The ordinary order statistics, progressively Type-II right censored order statistics and record values are considered as special cases. The results obtained in the article imply many results on exceedance statistics for the variety of models of ordered random variables.