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<p>The article investigates structural properties of the generalized Poisson (GenP) distribution and proposes a regression model for count data capable of handling high dispersion. The parameters are estimated via maximum likelihood and the Pearson residuals for the GenP regression model are studied. Simulation results show that the estimators exhibit good asymptotic behavior, and that the empirical distribution of residuals approximates the standard normal. In the application to fire occurrences in the Brazilian Amazon, this regression model outperforms the traditional log-linear model, as indicated by Pearson residual envelopes and a large estimated dispersion parameter, confirming strong overdispersion. For predictive evaluation, the GenP model is compared with Random Forest and XGBoost algorithms. XGBoost achieves the best predictive performance, with the highest <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(R^2\)</EquationSource> </InlineEquation> and some lowest measures. Although SHapley Additive exPlanations enable the exploration of variable effects in black-box models, they do not allow formal statistical inference as in regression models. Overall, the proposed regression model demonstrates solid asymptotic and residual properties, providing both interpretability and competitive predictive power compared to machine learning methods.</p>

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Generalized Poisson regression model: properties, residual analysis, and machine learning

  • 期刊论文

  • 发表日期：

    2025年12月16日

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  • Edwin M. M. Ortega,
  • Aldo Garay,
  • Gabriela M. Rodrigues,
  • Roberto Vila,
  • Gauss M. Cordeiro

摘要

The article investigates structural properties of the generalized Poisson (GenP) distribution and proposes a regression model for count data capable of handling high dispersion. The parameters are estimated via maximum likelihood and the Pearson residuals for the GenP regression model are studied. Simulation results show that the estimators exhibit good asymptotic behavior, and that the empirical distribution of residuals approximates the standard normal. In the application to fire occurrences in the Brazilian Amazon, this regression model outperforms the traditional log-linear model, as indicated by Pearson residual envelopes and a large estimated dispersion parameter, confirming strong overdispersion. For predictive evaluation, the GenP model is compared with Random Forest and XGBoost algorithms. XGBoost achieves the best predictive performance, with the highest \(R^2\) and some lowest measures. Although SHapley Additive exPlanations enable the exploration of variable effects in black-box models, they do not allow formal statistical inference as in regression models. Overall, the proposed regression model demonstrates solid asymptotic and residual properties, providing both interpretability and competitive predictive power compared to machine learning methods.