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