Good-bootstrap: simultaneous confidence intervals for large alphabet distributions

Simultaneous confidence intervals (SCI) for multinomial proportions are a corner stone in count data analysis and a key component in many applications. A variety of schemes were introduced over the years, mostly focussing on an asymptotic regime where the sample is large and the alphabet size is relatively small. In this work we introduce a new SCI framework which considers the large alphabet setup. Our proposed framework utilises bootstrap sampling with the Good-Turing probability estimator as a plug-in distribution. We demonstrate the favourable performance of our proposed method in synthetic and real-world experiments. Importantly, we provide an exact analytical expression for the bootstrapped statistic, which replaces the computationally costly sampling procedure. Our proposed framework is publicly available at the first author's Github page.