New Statistical Residuals for Regression Models in the Exponential Family: Characterization, Simulation, Computation, and Applications

Raydonal Ospina, Patrícia L. Espinheira, Leilo A. Arias, Cleber M. Xavier, Víctor Leiva () and Cecilia Castro
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Raydonal Ospina: Departamento de Estatística, CASTLab, Universidade Federal de Pernambuco, Recife 50670-901, Brazil
Patrícia L. Espinheira: Departamento de Estatística, CASTLab, Universidade Federal de Pernambuco, Recife 50670-901, Brazil
Leilo A. Arias: Departamento de Estatística, CASTLab, Universidade Federal de Pernambuco, Recife 50670-901, Brazil
Cleber M. Xavier: Departamento de Estatística e Ciências Atuariais, Universidade Federal de Sergipe, São Cristóvão 49107-230, Brazil
Víctor Leiva: School of Industrial Engineering, Pontificia Universidad Católica de Valparaíso, Valparaíso 2362807, Chile
Cecilia Castro: Centre of Mathematics, Universidade do Minho, 4710-057 Braga, Portugal

Mathematics, 2024, vol. 12, issue 20, 1-44

Abstract: Residuals are essential in regression analysis for evaluating model adequacy, validating assumptions, and detecting outliers or influential data. While traditional residuals perform well in linear regression, they face limitations in exponential family models, such as those based on the binomial and Poisson distributions, due to heteroscedasticity and dependence among observations. This article introduces a novel standardized combined residual for linear and nonlinear regression models within the exponential family. By integrating information from both the mean and dispersion sub-models, the new residual provides a unified diagnostic tool that enhances computational efficiency and eliminates the need for projection matrices. Simulation studies and real-world applications demonstrate its advantages in efficiency and interpretability over traditional residuals.

Keywords: advanced residual analysis; computational efficiency; exponential family models; Fisher scoring; mean and dispersion integration; model adequacy; regression diagnostics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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