Bounds for CDFs of Order Statistics Arising from INID Random Variables


Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.



In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a main concern for researchers. A cumulative distribution function (CDF) of these random variables (Fi:n) is a complex manipulating, long time consuming and a software-intensive tool that takes more and more times. Therefore, obtaining approximations and boundaries for Fi:n and other theoretical properties of these variables, such as moments, quantiles, characteristic function, and some related probabilities, has always been a main chal- lenge. Recently, Bayramoglu (2018) provided a new definition of ordering, by point to point ordering Fi’s (D-order) and showed that these new functions are CDFs and also, the corresponding random variables are independent. Thus, he suggested new CDFs (F[i]) that can be used as an alternative of Fi:n. Now with using, just F[1], and F[n], we have found the upper and lower bounds of Fi:n. Furthermore, specially a precisely approximation for F1:n and Fn:n (F1;n:n). Also in many cases approximations for other CDFs are derived. In addition, we compare approximated function with those o ered by Bayramoglu and it is shown that our results of these proposed functions are far better than D-order functions.


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Volume 19, Issue 1
June 2020
Pages 39-57
  • Receive Date: 23 July 2022
  • Revise Date: 29 September 2023
  • Accept Date: 23 July 2022