Beyond MSE in Poisson Ridge Regression: New Ridge Parameter Estimators with Additional Distributional Performance Criteria

Selman Mermi

doi:10.3390/math14071190

journal article Open Access Apr 02, 2026

Beyond MSE in Poisson Ridge Regression: New Ridge Parameter Estimators with Additional Distributional Performance Criteria

Selman Mermi

Mathematics Vol. 14 No. 7 pp. 1190 · MDPI AG

View at Publisher Save 10.3390/math14071190

Abstract

Despite its widespread use for mitigating multicollinearity in count data models, Poisson ridge regression (PRR) remains methodologically constrained by the choice of the ridge parameter k. Existing studies predominantly evaluate ridge parameter estimators using only the mean squared error (MSE) criterion, largely neglecting their distributional properties and estimation stability. Such a narrow evaluation framework may yield unreliable inference, particularly under high correlation and small sample sizes. This study makes two original contributions to the PRR literature. First, we conduct a comprehensive comparison of 13 commonly used ridge parameter estimators and introduce two new estimators that exhibit superior empirical performance. Second, we extend performance evaluation beyond MSE by incorporating outlier ratios and conformity to normality, thereby establishing a multidimensional framework that explicitly addresses distributional robustness and estimator stability. Monte Carlo simulations across 180 scenarios—varying the number of predictors, sample size, correlation level, and intercept value—show that several estimators deemed optimal under MSE perform poorly in terms of outlier prevalence and normality. In contrast, the proposed estimators consistently achieve a balanced performance between error minimization and distributional stability. Two real-data applications further support these findings.

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References

44

[1]

De Jong, P., and Heller, G.Z. (2008). Generalized Linear Models for Insurance Data, Cambridge University Press. 10.1017/cbo9780511755408

[2]

Osgood "Poisson-based regression analysis of aggregate crime rates" J. Quant. Criminol. (2000) 10.1023/a:1007521427059

[3]

Herawati "Poisson ridge regression for multicollinearity data: Case study of the number of maternal deaths in Lampung Province, Indonesia" Int. J. Innov. Res. Sci. Eng. Technol. (2024)

[4]

Frisch, R. (1934). Statistical Confluence Analysis by Means of Complete Regression Systems, University Institute of Economics.

[5]

Qasim "A new Poisson Liu regression estimator: Method and application" J. Appl. Stat. (2020) 10.1080/02664763.2019.1707485

[6]

Ridge Regression: Biased Estimation for Nonorthogonal Problems

Arthur E. Hoerl, Robert W. Kennard

Technometrics 1970 10.1080/00401706.1970.10488634

[7]

Kibria "Performance of some new ridge regression estimators" Commun. Stat.-Simul. Comput. (2003) 10.1081/sac-120017499

[8]

Muniz "On developing ridge regression parameters: A graphical investigation" Stat. Oper. Res. Trans. (2012)

[9]

Lukman "Review and classifications of the ridge parameter estimation techniques" Hacet. J. Math. Stat. (2017)

[10]

Kuvat "A new robust ridge parameter estimator based on search method for linear regression model" J. Appl. Stat. (2021) 10.1080/02664763.2020.1803814

[11]

Mansson "A Poisson ridge regression estimator" Econ. Model. (2011) 10.1016/j.econmod.2011.02.030

[12]

Kejian "A new class of biased estimate in linear regression" Commun. Stat.-Theory Methods (1993) 10.1080/03610929308831027

[13]

Mansson "Improved Liu estimators for the Poisson regression model" Int. J. Stat. Probab. (2012) 10.5539/ijsp.v1n1p2

[14]

"A new modified jackknifed estimator for the Poisson regression model" J. Appl. Stat. (2016) 10.1080/02664763.2015.1125861

[15]

Singh "An almost unbiased ridge estimator" Sankhyā Indian J. Stat. Ser. B (1986)

[16]

Rashad "A new ridge estimator for the Poisson regression model" Iran. J. Sci. Technol. Trans. A Sci. (2019) 10.1007/s40995-019-00769-3

[17]

Lukman "A new estimator for the multicollinear Poisson regression model: Simulation and application" Sci. Rep. (2021) 10.1038/s41598-021-82582-w

[18]

Kibria "A new ridge-type estimator for the linear regression model: Simulations and applications" Scientifica (2020) 10.1155/2020/9758378

[19]

Aladeitan "Modified Kibria–Lukman (MKL) estimator for the Poisson regression model: Application and simulation" F1000Research (2021) 10.12688/f1000research.53987.2

[20]

Yang "A new two-parameter estimator in linear regression" Commun. Stat.-Theory Methods (2010) 10.1080/03610920902807911

[21]

Asar "A new two-parameter estimator for the Poisson regression model" Iran. J. Sci. Technol. Trans. Sci. (2018) 10.1007/s40995-017-0174-4

[22]

Alkhateeb "Jackknifed Liu-type estimator in Poisson regression model" J. Iran. Stat. Soc. (2020) 10.29252/jirss.19.1.21

[23]

Lukman "Modified ridge-type for the Poisson regression model: Simulation and application" J. Appl. Stat. (2022) 10.1080/02664763.2021.1889998

[24]

Olakunle, K., Owolabi, A.T., and Olatayo, T.O. (2025). New Ridge-Type Estimator to Mitigate Multicollinearity in the Poisson Regression Model: Theory and Simulation, ZBW Leibniz Information Centre for Economics.

[25]

Kibria "A simulation study of some biasing parameters for the ridge type estimation of Poisson regression" Commun. Stat.-Simul. Comput. (2015) 10.1080/03610918.2013.796981

[26]

Mermi "How well do ridge parameter estimators perform in terms of normality, outlier detection, and MSE criteria?" Commun. Stat.-Simul. Comput. (2025) 10.1080/03610918.2024.2372078

[27]

Myers, R.H., Montgomery, D.C., Vining, G.G., and Robinson, T.J. (2010). Generalized Linear Models: With Applications in Engineering and the Sciences, Wiley. [2nd ed.]. 10.1002/9780470556986

[28]

Muniz "On some ridge regression estimators: An empirical comparison" Commun. Stat.-Simul. Comput. (2009) 10.1080/03610910802592838

[29]

Kibria "Some ridge regression estimators for the zero-inflated Poisson model" J. Appl. Stat. (2013) 10.1080/02664763.2012.752448

[30]

Dawoud "On the performance of the Poisson and the negative binomial ridge predictors" Commun. Stat.-Simul. Comput. (2018) 10.1080/03610918.2017.1324978

[31]

Hoerl "Ridge regression: Some simulations" Commun. Stat. (1975) 10.1080/03610927508827232

[32]

Algamal "Proposed methods in estimating the ridge regression parameter in Poisson regression model" Electron. J. Appl. Stat. Anal. (2018)

[33]

Oghenekevwe "Poisson ridge regression estimators: A performance test" Am. J. Theor. Appl. Stat. (2021) 10.11648/j.ajtas.20211002.13

[34]

Alkhamisi "Some modifications for choosing ridge parameters" Commun. Stat.-Theory Methods (2006) 10.1080/03610920600762905

[35]

Lukman "Classification-based ridge estimation techniques of Alkhamisi methods" J. Probab. Stat. Sci. (2018)

[36]

Ridge Regression: Applications to Nonorthogonal Problems

Arthur E. Hoerl, Robert W. Kennard

Technometrics 1970 10.1080/00401706.1970.10488635

[37]

Leys "Detecting outliers: Do not use standard deviation around the mean, use absolute deviation around the median" J. Exp. Soc. Psychol. (2013) 10.1016/j.jesp.2013.03.013

[38]

Statistical notes for clinical researchers: assessing normal distribution (2) using skewness and kurtosis

Hae-Young Kim

Restorative Dentistry & Endodontics 2013 10.5395/rde.2013.38.1.52

[39]

Hoyle, R.H. (1995). Structural equation models with nonnormal variables: Problems and remedies. Structural Equation Modeling, Sage Publications.

[40]

McDonald "A Monte Carlo evaluation of some ridge-type estimators" J. Am. Stat. Assoc. (1975) 10.1080/01621459.1975.10479882

[41]

Cameron, A.C., and Trivedi, P.K. (2013). Regression Analysis of Count Data, Cambridge University Press. [2nd ed.]. 10.1017/cbo9781139013567

[42]

Mansson "Estimating the unrestricted and restricted Liu estimators for the Poisson regression model: Method and application" Comput. Econ. (2020) 10.1007/s10614-020-10028-y

[43]

Abonazel "Developing robust ridge estimators for Poisson regression model" Concurr. Comput. Pract. Exp. (2022) 10.1002/cpe.6979

[44]

(2026, March 24). Cengage. Available online: https://www.cengage.com/cgi-wadsworth/course_products_wp.pl?fid=M20b&product_isbn_issn=9781111531041.

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Published: Apr 02, 2026
Vol/Issue: 14(7)
Pages: 1190
License: View

Authors

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Selman Mermi

Department of Statistics, Faculty of Science, Muğla Sıtkı Koçman University, Muğla 48000, Türkiye

Cite This Article

Selman Mermi (2026). Beyond MSE in Poisson Ridge Regression: New Ridge Parameter Estimators with Additional Distributional Performance Criteria. Mathematics, 14(7), 1190. https://doi.org/10.3390/math14071190

Beyond MSE in Poisson Ridge Regression: New Ridge Parameter Estimators with Additional Distributional Performance Criteria

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