Bounds for CDFs of Order Statistics Arising from INID Random Variables

In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a remarkable concern for researchers. The cumulative distribution function (CDF) of these random variables (Fi:n) is a complex manipulating, long time consuming and a softwareintensive tool that takes considerable time. Therefore, obtain approximations and bounds for Fi:n and other theoretical properties of these variables, such as moments, quantiles, characteristic functions, and some related probabilities, has always been the main challenge. Recently, Bayramoglu (2018), Bayramoglu (2018), has introduced a set of CDFs (F[i]), whose definitions are based on a point to point ordering of the original CDFs (Fi), that can be used to approximate the CDF of i-th order statistics (Fi:n). Here, by using just F[1] and F[n], we provide new upper and lower bounds for the Fi:n. Furthermore, new approximations for F1:n and Fn:n, as well as for other cases, are derived. Comparisons with respect to approximations suggested by Bayramoglu Bayramoglu (2018) are also provided.