Fuzzy Improved Distributions for Exceedance Counts in Order Statistic Intervals

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 Fμ. 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=Fμ). 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.