Distribution of the Number of Observations Greater Than the ith Dependent Progressively Type-II Censored Order Statistic and Its Use in Goodness-of-fit Testing

In a life-testing problem, it may be of interest to investigate the number of observations close to but greater than the median, minimum, or more generally, the ith progressively Type-II censored order statistic (PCOS-II). In this paper, we derive the probability mass and distribution functions of the number of observations greater than the ith PCOS-II for a system with identical components for the cases of independent as well as dependent components. The type of dependence considered among the component lifetimes is through an Archimedean copula. We also provide a goodness-of-fit method for determining the best copula model for a given PCOS-II. Finally, an example is provided to illustrate the results developed here.     

Order statistics and test duration

There are two principal issues in statistical planning. One is the accuracy/reliability of statistical inference and the other is the length of test time needed to complete the designed experiment. With regard to the latter, various test schemes have been proposed and applied in statistical literature. These schemes, among others, include type-I censoring, the usual type-II censoring, and progressively type-II censoring. To implement any of these experiments it is necessary that the capacity of the test facility is large enough so that all the items can be tested simultaneously. If, however, instead of having one facility with large capacity there are several facilities with relatively smaller capacities, a differently designed experiment would be necessary. This paper studies and compares elapsed test times and total elapsed test times corresponding to different statistical plans. The results obtained here are useful for performing an experiment that has shorter test time in a certain sense.