A Dawoud–Kibria Shrinkage Estimator for the Waring Regression Model: Methodology, Simulation, and Application

Abstract

Count regression models often suffer from two concurrent problems: overdispersion in the response and multicollinearity among predictors, which together destabilize maximum likelihood estimation in the Waring regression model (WRM). To address this issue, we extend the Dawoud--Kibria two‑parameter shrinkage strategy to the WRM and develop a new estimator, termed the Waring Dawoud--Kibria estimator (WRDKE). The proposed estimator introduces two biasing parameters into the WRM score equations and yields a closed‑form solution whose bias and variance are characterized analytically via the matrix mean squared error (MMSE) and scalar mean squared error (MSE). We derive conditions under which the WRDKE dominates the classical maximum likelihood estimator (MLE), the Waring ridge regression estimator (WRRE), and the Waring Liu estimator (WRLE) in the MMSE sense. A comprehensive Monte Carlo simulation study is conducted under varying sample sizes, numbers of predictors, pairwise correlations, and overdispersion levels. Predictors are generated following the McDonald--Galarneau scheme to induce controlled multicollinearity, while responses are drawn from the Waring distribution with different overdispersion parameters. Across all configurations, the WRDKE consistently achieves the smallest average MSE (AMSE), with reductions in AMSE relative to MLE that exceed 50% in severe multicollinearity scenarios and remain substantial even for moderate correlations and larger sample sizes. The WRDKE also outperforms WRRE and WRLE uniformly, yielding notably lower AMSE values such as 2.05 versus 5.28 for MLE when the sample size is 500 and correlation is 0.85. Finally, an application to real overdispersed count data illustrates that the WRDKE provides more stable coefficient estimates and improved in‑sample fit compared with existing WRM estimators.