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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.

Consider an infinite sequence of independent identically distributed (i.i.d.) continuous random variables. We begin with an initial set of m random variables and keep in our list only the top m values as new variables are observed sequentially and added to the data set. We obtain the joint distribution of the ordered observations in our list after the lth change occurs to the list. We also obtain the probability of a change in the list given that there are already l changes. We discuss the relationship between the proposed model, ordinary order statistics, and m-records.

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.