Minimax and adaptive tests for detecting abrupt and possibly transitory changes in a Poisson process

Motivated by applications in cybersecurity and epidemiology, we consider the of detecting an abrupt change in the intensity of a Poisson process, by a jump (non transitory change) or a bump (transitory change) constant. We propose a complete study from the nonasymptotic minimax point of view, when the constant baseline intensity is known or. The question of minimax adaptation with respect to each parameter(height, location, length) of the change is tackled, leading to a comprehensive of the various minimax separation rate regimes. We exhibit three such and identify the factors of the two phase transitions, by giving the of adaptation to each parameter. For each alternative hypothesis, on the knowledge or not of each change parameter, we propose minimax minimax adaptive tests based on linear statistics, close to CUSUM, or quadratic statistics more adapted to the L 2-distance considered our minimax criteria and typically more powerful in practice, as our study shows. When the change location or length is unknown, our tests are constructed from a scan aggregation principle combined with or min-p level correction, and a conditioning trick when the intensity is unknown.