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

Abstract

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. Starting from the totally agnostic case where all the parameters (height, location, length) of the change are unknown, the question of minimax adaptation with respect to each parameter 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 any alternative hypothesis, depending on the knowledge or not of the change parameters, we propose minimax or minimax adaptive tests based on simple linear counting statistics derived from Neyman-Pearson tests, or quadratic statistics. When the change location or length is unknown, minimax adaptation is obtained from a scan aggregation principle combined with a Bonferroni or min- p level correction, and a conditioning trick when the baseline intensity is unknown. MSC2020 subject classifications: 62C20, 62M20, 62F03, 62G10, 60G55, 60G25, 60G35, 60E15, 62E17.