International Journal of System Assurance Engineering and Management Pub Date : 2021-03-07 , DOI: 10.1007/s13198-021-01078-y S. Nadarajah , I. E. Okorie
Given a random sample \(X_1, X_2, \ldots , X_n\), the distributions of \(\min \left( X_1, X_2, \ldots , X_n \right)\) and \(\max \left( X_1, X_2, \ldots , X_n \right)\) are of interest in many areas. We derive explicit expressions for moments of \(\min \left( X_1, X_2, \ldots , X_n \right)\) and \(\max \left( X_1, X_2, \ldots , X_n \right)\) for thirty four families of distributions, including the normal and Student’s t distributions. These results can be especially useful when data are scarce. The correctness of the expressions is checked by a simulation study. Applications to two engineering data sets are given.
中文翻译:
关于单变量分布的最大值和最小值
给定随机样本\(X_1,X_2,\ ldots,X_n \),\(\ min \ left(X_1,X_2,\ ldots,X_n \ right)\)和\(\ max \ left(X_1, X_2,\ ldots,X_n \ right)\)在许多领域都很受关注。我们导出三十分钟\(\ min \ left(X_1,X_2,\ ldots,X_n \ right)\)和\(\ max \ left(X_1,X_2,\ ldots,X_n \ right)\)的显式表达式四个分布族,包括正态分布和学生t分布。当数据稀缺时,这些结果可能特别有用。表达式的正确性通过模拟研究进行检查。给出了对两个工程数据集的应用。