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Anytime-Valid Continuous-Time Confidence Processes for Inhomogeneous Poisson Processes

11 October 2024
Michael Lindon
Nathan Kallus
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Abstract

Motivated by monitoring the arrival of incoming adverse events such as customer support calls or crash reports from users exposed to an experimental product change, we consider sequential hypothesis testing of continuous-time inhomogeneous Poisson point processes. Specifically, we provide an interval-valued confidence process Cα(t)C^\alpha(t)Cα(t) over continuous time ttt for the cumulative arrival rate Λ(t)=∫0tλ(s)ds\Lambda(t) = \int_0^t \lambda(s) \mathrm{d}sΛ(t)=∫0t​λ(s)ds with a continuous-time anytime-valid coverage guarantee P[Λ(t)∈Cα(t) ∀t>0]≥1−α\mathbb{P}[\Lambda(t) \in C^\alpha(t) \, \forall t >0] \geq 1-\alphaP[Λ(t)∈Cα(t)∀t>0]≥1−α. We extend our results to compare two independent arrival processes by constructing multivariate confidence processes and a closed-form eee-process for testing the equality of rates with a time-uniform Type-I error guarantee at a nominal α\alphaα. We characterize the asymptotic growth rate of the proposed eee-process under the alternative and show that it has power 1 when the average rates of the two Poisson process differ in the limit. We also observe a complementary relationship between our multivariate confidence process and the universal inference eee-process for testing composite null hypotheses.

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