A Systematic Framework for Composite Hypothesis Testing of Independent Bernoulli Trials

This letter is focused on the classic problem of testing samples drawn from independent Bernoulli probability mass functions, when the success probability under the alternative hypothesis is not known. The goal is to provide a systematic taxonomy of the viable detectors (designed according to theoretically-founded criteria) which can be used for the specific instance of the problem. Both One-Sided (OS) and Two-Sided (TS) tests are considered, with reference to: (i) identical success probability (a homogeneous scenario) or (ii) different success probabilities (a non-homogeneous scenario) for the observed samples. As a result of the study, a complete summary (in tabular form) of the relevant statistics for the problem is provided, along with a discussion on the existence of the Uniformly Most Powerful (UMP) test. Finally, when the Likelihood Ratio Test (LRT) is not UMP, existence of the UMP detector after reduction by invariance is investigated.