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

TL;DR

This paper develops anytime-valid confidence processes for inhomogeneous Poisson processes in continuous time, enabling sequential hypothesis testing with uniform error guarantees and extending to compare multiple processes.

Contribution

It introduces a novel interval-valued confidence process for the cumulative rate and constructs multivariate confidence processes and $e$-processes for rate comparison with uniform error control.

Findings

Provides continuous-time confidence processes with coverage guarantees

Constructs an $e$-process for testing rate equality with power 1 asymptotically

Characterizes the growth rate of the $e$-process under alternatives

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 $C^\alpha(t)$ over continuous time $t$ for the cumulative arrival rate $\Lambda(t) = \int_0^t \lambda(s) \mathrm{d}s$ with a continuous-time anytime-valid coverage guarantee $\mathbb{P}[\Lambda(t) \in C^\alpha(t) \, \forall t >0] \geq 1-\alpha$. We extend our results to compare two independent arrival processes by constructing multivariate confidence processes and a closed-form $e$-process for testing the equality of rates with a time-uniform Type-I error guarantee at a nominal $\alpha$. We characterize the asymptotic growth rate of the proposed $e$-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 $e$-process for testing composite null hypotheses.