Zero-count Detector.

Area of Science:

• Statistical theory
• Rare event analysis
• Physics and Health Physics applications

Background:

• Zero-count detection (ZCD) is prevalent in fields counting rare events.
• Classical statistics offers no adequate solutions for ZCD.
• Bayesian statistics is necessary for addressing ZCD challenges.

Purpose of the Study:

• To develop a statistical theory for zero-count detection (ZCD).
• To explore Bayesian approaches for ZCD, including prior selection and estimation.
• To evaluate alternative distributions like zero-inflated Poisson and Negative-binomial for ZCD.

Main Methods:

• Development of a statistical theory for zero-class Poisson under specific conditions.
• Investigation of uniform and reference priors in Bayesian statistics.
• Derivation of Bayesian posteriors, point estimates, and upper limits.
• Analysis of zero-inflated Poisson and Negative-binomial distributions using Bayesian marginalization.

Main Results:

• The maximum-entropy prior demonstrated the least bias and risk, proving most admissible for ZCD.
• Bayesian methods provide acceptable solutions where classical statistics fail for ZCD.
• Zero-inflated Poisson and Negative-binomial distributions simplify to Poisson under limited information via Bayesian marginalization.

Conclusions:

• A robust statistical theory for ZCD has been established using Bayesian inference.
• The maximum-entropy prior is recommended for ZCD analysis due to its optimal properties.
• Bayesian marginalization clarifies the relationship between advanced distributions and the Poisson distribution in ZCD contexts.