Abstract
Ranked set sampling (RSS) helps when ranking units is cheap but measuring them is expensive, because it lets you measure only the units whose ranks tell you the most. This paper looks at a variant called except extreme ranked set sampling (EERSS), which drops the smallest and largest unit from every set and keeps only the middle ranks, and asks how well it estimates the cumulative distribution function F(x). We first derive the plain empirical CDF estimator under EERSS, along with its exact bias, mean squared error, and efficiency relative to simple random sampling (SRS). Unlike RSS, this estimator is not unbiased in general; it is exactly unbiased only at the point where F(x)=1/2, no matter what the population looks like. To fix this, we build a maximum likelihood estimator instead, treating each indicator of whether a ranked unit falls below x as a binomial outcome with a known probability tied to its rank. This estimator needs no assumption about the population's shape, and its Fisher information gives both an asymptotic variance and a confidence interval for F(x). A numerical comparison then puts both versions of EERSS side by side with SRS and RSS, in terms of bias, MSE, and relative efficiency, across several sample sizes and values of F(x). The clearest finding is this: the maximum likelihood EERSS estimator is the best choice near the centre of the distribution, clearly beating RSS and SRS there, while RSS remains the better choice in the tails. The plain empirical EERSS estimator, on the other hand, should generally be avoided, since its bias never shrinks no matter how much data is collected; switching to maximum likelihood is what actually makes EERSS worth using. A real dataset of combined cycle power plant measurements confirms that this pattern holds in practice, not just in theory.