Maximum-likelihood regression with systematic errors for astronomy and the physical sciences: I. Methodology and goodness-of-fit statistic of Poisson data

TL;DR

This paper introduces a new statistical method for incorporating systematic errors into maximum-likelihood regression of Poisson data, improving goodness-of-fit assessments in astronomy and physical sciences.

Contribution

It develops a quasi-maximum-likelihood approach that generalizes the Poisson deviance to account for systematic errors, with theoretical, simulation, and real-data validation.

Findings

The generalized deviance effectively accounts for over-dispersion due to systematic errors.

The method is applicable to a wide range of integer-count data analysis scenarios.

Validation shows improved fit assessment in astronomical data.

Abstract

The paper presents a new statistical method that enables the use of systematic errors in the maximum-likelihood regression of integer-count Poisson data to a parametric model. The method is primarily aimed at the characterization of the goodness-of-fit statistic in the presence of the over-dispersion that is induced by sources of systematic error, and is based on a quasi-maximum-likelihood method that retains the Poisson distribution of the data. We show that the Poisson deviance, which is the usual goodness-of-fit statistic and that is commonly referred to in astronomy as the Cash statistics, can be easily generalized in the presence of systematic errors, under rather general conditions. The method and the associated statistics are first developed theoretically, and then they are tested with the aid of numerical simulations and further illustrated with real-life data from astronomical observations. The statistical methods presented in this paper are intended as a simple general-purpose framework to include additional sources of uncertainty for the analysis of integer-count data in a variety of practical data analysis situations.