Fuzzy Improved Distributions for Exceedance Counts in Order Statistic Intervals

Abstract

We study exceedance counts for order statistic intervals when boundary uncertainty is modeled through a fuzzy improved distribution function. In an ordinary setting, whether an observation falls below a threshold is decided by a crisp comparison, which can be unstable when specifications are vague, subject to tolerance bands, or expressed linguistically. We replace the crisp rule by a graded membership function and use the fuzzy improved cumulative distribution function F mu. From an initial independent and identically distributed sample, with ordinary cumulative distribution function F, we form the random interval between the r-th and s-th order statistics, and we count how many of m independent newcomers fall inside this interval. Newcomers follow either the ordinary model (Q=F) or the fuzzy improved model (Q=F mu). We derive exact finite-sample formulas, moments, and a distribution-free representation based on a probability integral transform, which yields the large-m limit law of the newcomer proportion. Numerical illustrations for exponential and uniform distributions show how fuzzification reshapes the distribution and can materially change predictive dispersion of exceedance counts.