Nonparametric confidence intervals and tolerance limits based on minima and maxima

Let X-i,X-j (1 <= i <= k, 1 <= j <= n(i)) be independent random variables and for a fixed i, X-i,X-j's, (1 <= j <= n(i)) be identically distributed random variables with survival function (F) over bar (i) = (F) over bar (alpha i), where alpha(i) is a known positive constant. Also, suppose M-i and M-i', respectively, denote the maximum and minimum of the ith sample. This article investigates the nonparametric confidence intervals for an arbitrary quantile of the distribution F and tolerance limits based on these statistics. Various cases have been studied and in each case, the nonparametric confidence intervals are obtained and exact expressions for the confidence coefficients of these confidence intervals are derived. A data set representing the time of successive failures of the air conditioning system on Boeing 720 jet aircraft is used to illustrate the results. Finally, the accuracy of the proposed procedure has been investigated, when alpha(i)'s are unknown via a simulation study.