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On the maximum and minimum for classes of univariate distributions

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Abstract

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.

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Acknowledgements

The authors would like to thank the Editor and the three referees for careful reading and comments which greatly improved the paper.

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  1. Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK

    S. Nadarajah & I. E. Okorie

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  1. S. Nadarajah
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  2. I. E. Okorie
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Correspondence to S. Nadarajah.

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Nadarajah, S., Okorie, I.E. On the maximum and minimum for classes of univariate distributions. Int J Syst Assur Eng Manag 12, 290–309 (2021). https://doi.org/10.1007/s13198-021-01078-y

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  • DOI: https://doi.org/10.1007/s13198-021-01078-y

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