Multi-Threshold Sampling: Local and Global Parameter Identifiability

Abstract

Multi-threshold (MT) sampling records threshold-crossing times for parameter estimation in applications including positron emission tomography, oil well logging, and photon-counting X-ray imaging. We study whether the distribution of recorded crossings uniquely determines the signal parameters and how many thresholds are required for this identifiability. For a fixed MT sampler, we show that the ranks of distribution derivatives with respect to the parameters cannot increase through crossing, timing, and selection. Under suitable regularity assumptions, we characterize local identifiability by a full-rank Jacobian of finitely many event probabilities defined on recorded crossings and establish a necessary and sufficient global identifiability criterion under additional rank and topological assumptions. For a nonhomogeneous Poisson signal model with four intensity parameters, a fixed known single-photon response, and deterministic MT sampling, we prove that four cumulative probabilities of the first recorded crossing time establish local and global identifiability under their respective conditions using a single sufficiently small positive threshold. This attains the minimum number of thresholds for the specified model and sampler. Numerical experiments illustrate these results and reveal weak parameter sensitivity and substantial finite-sample estimation errors when estimation uses these probabilities. The results establish parameter uniqueness while highlighting the role of statistical sensitivity in accurate estimation.