article · Scientific Reports
A new statistical estimator, the Poisson-modified quasi-Lindley improved Liu-type estimator, addresses challenges in analysing overdispersed count data when explanatory variables are highly correlated. The method incorporates an eigenvalue-dependent biasing function, creating a flexible one-parameter shrinkage scheme that tailors shrinkage to each regression direction. Theoretical derivations establish closed-form expressions for its bias, variance-covariance matrix, and mean squared error. Comprehensive Monte Carlo simulations evaluating various sample sizes, predictor counts, dispersion levels, and collinearity degrees demonstrate that the technique uniformly outperforms standard maximum likelihood estimation and existing shrinkage methods in terms of mean squared error. These performance gains are particularly notable under severe multicollinearity and moderate-to-large sample sizes. An empirical evaluation using Swedish football league data confirms that the estimator stabilises regression coefficients without changing their signs or relative rankings, offering an efficient approach for complex count data analysis.
Analysing count data often becomes unreliable when predictive variables are closely correlated, leading to unstable estimates. By effectively reducing estimation error while preserving the directional meaning of predictors, this mathematical advancement provides researchers with a more accurate tool for interpreting complex datasets characterised by overdispersion and multicollinearity, ensuring that statistical models yield dependable and interpretable findings.
The methodology can be integrated into statistical software packages, analytics platforms, or sports analytics workflows that process overdispersed count metrics. Potential end users include data scientists, quantitative analysts, and sports researchers working with collinear datasets. Because the research establishes the mathematical foundation and validates it through simulation and a real-world sports dataset, the technique is at an applied and tested stage, ready for algorithmic implementation in computational libraries.
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Abstract This study develops an improved Liu‑type estimator for the Poisson‑modification of the quasi‑Lindley regression model (PMQL‑RM) to handle overdispersed count data in the presence of severe multicollinearity. We first formulate the PMQL‑RM and review existing shrinkage estimators, including the PMQL ridge estimator, the PMQL Liu estimator, and a previously proposed PMQL Liu‑type estimator. Building on recent work on improved Liu‑type estimators in generalized linear models, we introduce a new PMQL improved Liu‑type estimator (PMQL‑ILTE) that incorporates an eigenvalue‑dependent biasing function, yielding a flexible one‑parameter shrinkage scheme so that the shrinkage applied to each regression direction is tailored to the corresponding eigenvalue of the information matrix. We derive closed‑form expressions for the bias, variance–covariance matrix, matrix mean squared error (MMSE), and scalar MSE. A comprehensive Monte Carlo study is then conducted, varying multicollinearity levels, sample sizes, number of predictors, and dispersion parameters. The simulation results show that the PMQL‑ILTE uniformly achieves the smallest MSE across all scenarios, with especially pronounced gains under high multicollinearity and moderate‑to‑large samples. Finally, an application to Swedish football league data demonstrates the practical relevance of the proposed estimator. Relative to PMQL‑MLE and other shrinkage competitors, PMQL‑ILTE substantially reduces the estimated MSE while stabilizing regression coefficients without altering their signs or substantive ranking. Overall, the results indicate that the proposed estimator provides an efficient alternative for modeling overdispersed count responses with multicollinear covariates.
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DOI: 10.1038/s41598-026-69426-1
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