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.

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Assume that only the lists of upper k-records and lower k-records of a …nite sequence are available and the existence of a monotonic trend in location is interested in. In this study, a distribution-free test based on the di¤erence between the numbers of upper and lower k-records is proposed for this situation. The exact and asymptotic distributions of the proposed test statistic are obtained for a random continuous sequence which is independent and identically distributed (i.i.d.). Also, a comparison between the proposed test and some well-known distribution-free tests is made in terms of empirical powers.