Minimax and adaptive test for detecting a transitory change in a Poisson process

Considering a Poisson process observed on a bounded, fixed interval, we are interested in the problem of detecting a transitory change in its distribution, characterised by a bump in its intensity. Formulated as an off-line change-point problem, this study aims at proposing a non-asymptotic minimax testing set-up to construct a minimax and adaptive detection procedure. In particular, we establish the minimax separation rates over various classes of alternatives, defined according to whether or not the jump position, length and/or size are known.