Minimax and adaptive tests for detecting an abrupt change in a Poisson process

We adress the question of detecting a single change point characterized by a jump in the intensity of a Poisson process, defined with respect to some measure $Ldt$ ($L>0$) over the interval $[0,1]$. This work, which can be viewed as a preliminary step to the construction of change point localization procedures, presents a nonasymptotic minimax study. By considering the usual distance of $\mathbb{L}^2([0,1])$, we establish the minimax separation rates over various classes of alternatives, defined according to whether or not the jump position and/or size are known.

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