Fundamental Bounds and Efficient Estimation for Dead-Time-Constrained Event Detection, with Application to Single-Photon Lidar

Technical Summary: Fundamental Bounds and Efficient Estimation for Dead-Time-Constrained Event Detection

Problem Formulation
The paper addresses the statistical problem of estimating physical parameters from periodic binary event-detection processes subject to nonparalyzable dead time and gating. In such systems, a detection event renders the detector inactive for a fixed recovery interval $t_d$ (dead time), during which no further events can be registered. Additionally, the experimenter may employ gating, a design choice where the detector is only activated during specific time bins to optimize information gathering or mitigate pile-up.

This setup creates a non-independent and identically distributed (non-i.i.d.) stochastic process. The authors formalize this class of problems as Dead-Time Event Detection (DED) processes. The core challenge is that dead time and gating introduce a systematic discrepancy between observed detection statistics and the underlying physical event rates, rendering standard i.i.d. asymptotic theories inapplicable. The goal is to determine the fundamental lower bounds for estimation accuracy in this setting and to identify estimators that achieve these bounds.

Methodology
The authors develop an asymptotic statistical theory for DED processes as the number of observation periods $T \to \infty$. The methodology proceeds through the following steps:

1. Model Definition: The process is defined by periodic incidence probabilities $p_t(\theta)$ (where $\theta$ is the parameter of interest) and a gating rule that determines detector activation $G_t$. The observed outcome is $Y_t = G_t Z_t$, where $Z_t$ is the latent Bernoulli event. The model enforces causality and the dead-time constraint: a detection at time $t$ forces $G_s = 0$ for $s \in \{t+1, \dots, t+D\}$.
2. Sufficient Statistic Identification: Using the Neyman–Fisher factorization theorem, the authors derive the likelihood function for the process. They identify a sufficient statistic consisting of phasewise aggregate counts: the number of active bins ($N_r$) and the number of detections ($S_r$) for each phase $r$ within the period $K$. This result implies that hardware histogramming, which typically discards $N_r$ (retaining only $S_r$), may lose statistically useful information, particularly for adaptive gating schemes.
3. Local Asymptotic Normality (LAN): To handle the non-i.i.d. nature of the data, the authors establish that DED processes satisfy the Local Asymptotic Normality (LAN) property. They derive the Fisher information rate $I(\theta; \gamma)$, where $\gamma$ represents the limiting gating frequencies. The information rate is shown to be a weighted sum of the standard Bernoulli information for each phase, weighted by how often that phase is sampled.
4. Lower Bounds: Using the LAN framework, the authors apply the Hájek convolution theorem and the local asymptotic minimax theorem to derive fundamental lower bounds on the asymptotic variance of any regular estimator. These bounds are characterized by the inverse of the Fisher information rate, $I_0^{-1}$.
5. Estimator Construction:
   • Maximum Likelihood Estimator (MLE): The authors prove that the MLE, which maximizes the derived likelihood, is asymptotically efficient, attaining the derived lower bounds.
   • One-Step Estimators: Recognizing that computing the MLE often requires solving a nonconvex optimization problem, the authors propose Le Cam one-step estimators. These estimators start with a "pilot" estimator (which need only be $\sqrt{T}$-consistent, not fully efficient) and apply a single Newton-Raphson correction step using the score function and the estimated Fisher information.

Key Contributions

• Theoretical Framework: The paper provides the first rigorous asymptotic theory for parameter estimation in periodic binary processes with dead time and arbitrary gating. It establishes the LAN property and derives the corresponding Fisher information rate for this specific non-i.i.d. setting.
• Fundamental Bounds: It quantifies the fundamental limits of estimation accuracy, demonstrating that gating rules affect asymptotic information solely through their limiting sampling frequencies.
• Sufficient Statistic: It identifies that the count of active detector bins ($N_r$) is a necessary component of the sufficient statistic, challenging the common practice of discarding this data in hardware histogramming.
• Efficient Estimators: The paper proves the asymptotic efficiency of the MLE for general gating rules and introduces a class of one-step estimators that achieve the same efficiency with significantly lower computational cost (avoiding iterative nonconvex optimization).
• Robust Pilot Design: The authors introduce specific pilot estimators for lidar applications, including a robust matched-filter estimator that uses median-based background subtraction and template matching, offering stability against model misspecification and initialization sensitivity.

Results and Validation
The theoretical results are validated through simulations and real-data experiments in the context of single-photon lidar:

• Asymptotic Efficiency: Simulations confirm that both the MLE and the one-step estimators converge to the derived Fisher-information lower bound as the observation time increases.
• Impact of Dead Time: The study demonstrates that neglecting dead time leads to a substantial underestimation of the minimum attainable error (by approximately 75% in high-flux scenarios).
• Initialization Sensitivity: The MLE is shown to be sensitive to initialization; poor initial guesses can lead to convergence to local optima, especially at moderate sample sizes. The one-step estimator, when paired with a robust pilot, avoids this pitfall while maintaining efficiency.
• Robustness: Under model misspecification (e.g., unmodeled background bumps), the robust pilot combined with a one-step update outperforms standard MLE approaches initialized with less robust methods (like Fourier inversion).
• Real-Data Performance: Experiments on real single-photon lidar data confirm that the asymptotic theory remains predictive at practical sample sizes and that the one-step estimator effectively corrects biases in existing literature estimators.

Significance
The paper claims significance in filling a gap in statistical theory for dead-time-constrained systems, which are prevalent in fields such as lidar, fluorescence lifetime imaging, X-ray astronomy, and nuclear physics. By providing a rigorous framework for fundamental bounds, the work guides the principled design of gating schemes and data compression strategies (via the identification of sufficient statistics). Furthermore, the proposal of one-step estimators offers a practical alternative to computationally expensive MLE optimization, enabling efficient and robust parameter estimation in real-world applications where model misspecification and initialization sensitivity are critical concerns. The authors note that their theory can inform future hardware design and data compression in bandwidth-limited systems.