Optimal Intervals for Fisher’s problem of the Nile

Fisher’s Nile problem, a theoretical framework originally introduced to illustrate the use of ancillary statistics for parameter estimation, has intrigued statisticians for decades. The challenges of constructing optimal interval estimators in both the original problem and scenarios with unequal sample sizes remain largely unexplored. This article bridges this gap by developing scale-equivariant confidence intervals with the shortest expected length for a given confidence level. This optimal interval is derived from the Bayes posterior distribution from the scale prior, but with non constant credibility. We provide a general way of constructing pivotal intervals with given coverage. We prove that this pivotal construction also gives the optimal interval. This simplifies its computation. These results offer potential for further exploration of equivariant interval estimation in real-world scenarios.