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‎The equilibrium distributions have many applications in reliability theory, stochastic orderings and random processes. ‎The purpose of this paper is to introduce the equilibrium distributions and presents some results related to this issue. Some results are based on order statistics. ‎In this paper, ‎the generalized Pareto distributions are also analyzed and some basic relationships between the equilibrium distributions are presented.

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In many quality control applications, the necessary distributional assumptions to correctly apply the traditional parametric control charts are either not met or there is simply not enough information or evidence to verify the assumptions. It is well known that performance of many parametric control charts can be seriously degraded in situations like this. Thus, control charts that do not require a specific distributional assumption to be valid, so-called nonparametric or distribution-free charts, are desirable in practice. In this paper, a simple to use multivariate nonparametric control chart is introduced. The chart is based on the multivariate two sample Mann-Withney Wilcoxon test for equality of location vectors of two populations. Using simulated data we show that there are situations in which the Mann-Withney multivariate control chart has a better performance compared with T2 control chart.