Confidence sets for the maximizers of intensity functions

Identifying times or time intervals when the intensity function of a Poisson process is maximal is important in a variety of practical problems, for instance in traffic control or with planning issues involving customer arrivals or accident occurrences. For this purpose, we propose confidence sets that are intuitive and easy to obtain, which makes them practicable for a quick exploratory data analysis. They may also be used in the context of mode estimation for probability densities. In the current literature, confidence sets for the mode are based on the assumption of an-at least locally-unique mode. In contrast, our approach retains the coverage probability even if the underlying intensity or density has a flat top. We even allow the intensity to be constant in the extreme.