The beta regression model (BRM) is widely used for analyzing bounded response variables, such as proportions, percentages. However, when multicollinearity exists among explanatory variables, the conventional maximum likelihood estimator (MLE) becomes unstable and inefficient. To address this issue, we propose new modified Liu estimators for the BRM, designed to enhance estimation accuracy in the presence of high multicollinearity among predictors. The proposed estimators extend the traditional Liu estimator by incorporating flexible biasing parameters, offering a more robust alternative to the MLE. Theoretical comparisons demonstrate the superiority of the new estimators over existing methods. Additionally, Monte Carlo simulations and real-world applications evidence their improved performance in terms of mean squared error (MSE) and mean absolute error (MAE). The results indicate that the proposed estimators significantly reduce estimation bias and variance under multicollinearity, providing more reliable regression coefficients.

This study introduces a new two-parameter Liu estimator (PMTPLE) for addressing the multicollinearity problem in the Poisson regression model (PRM). The estimation of the PRM is traditionally accomplished through the Poisson maximum likelihood estimator (PMLE). However, when the explanatory variables are correlated, thus leading to multicollinearity, the variance or standard error of the PMLE is inflated. To address this issue, several alternative estimators have been introduced, including the Poisson ridge regression estimator (PRRE), Liu estimator (PLE), and adjusted Liu estimator (PALE), each of them relying on a single shrinkage parameter. The PMTPLE uses two shrinkage parameters, which enhances its adaptability and robustness in the presence of multicollinearity between explanatory variables. To assess the performance of the PMTPLE compared to the four existing estimators (the PMLE, PRRE, PLE, and PALE), a simulation study is conducted that encompasses various scenarios and two empirical applications. The evaluation of the performance is based on the mean square error (MSE) criterion. The theoretical comparison, simulation results, and findings of the two applications consistently demonstrate the superiority of the PMTPLE over the other estimators, establishing it as a robust solution for count data analysis under multicollinearity conditions.

The Poisson maximum likelihood estimator (PMLE) is commonly used to estimate the coefficients of the Poisson regression model (PRM). However, it is well known that the PMLE is highly sensitive to outliers, which can distort the estimated coefficients and lead to misleading results. Different studies have provided robust Poisson regression estimators to alleviate this problem. Additionally, the PMLE is sensitive to multicollinearity, since multicollinearity leads to an inflation of the variance, an error in the signals of the coefficients, as well as an increase in the mean squared error (MSE) value. Therefore, several biased Poisson estimators have been provided to handle this problem, such as the Poisson ridge estimator and the Poisson modified ridge-type estimator. Despite different Poisson-biased regression estimators being proposed, there has been no analysis of the robust version of these estimators to deal with the two above-mentioned problems simultaneously, except for minimal work. This paper proposed a new robust Poisson two-parameter estimator (PMT-PTE), which combines the transformed M-estimator (MT) with the two-parameter estimator, providing a new approach to handling outliers and multicollinearity. Theoretical comparisons and Monte Carlo simulations were conducted to show the proposed performance compared with the other estimators. The simulation results indicated that the proposed PMT-PTE estimator outperformed the other estimators in different scenarios, in cases where both problems existed. Finally, we analyzed a real-world dataset that encompasses both problems, and the results confirmed the theoretical and simulation results and demonstrated the superiority of the proposed estimator, introducing a new approach to data analysis.

The Poisson-Inverse Gaussian regression model is a widely used method for analyzing count data, particularly in over-dispersion. However, the reliability of parameter estimates obtained through maximum likelihood estimation in this model can be compromised when multicollinearity exists among the explanatory variables. Multicollinearity means that high correlations between explanatory variables inflate the variance of the maximum likelihood estimates and increase the mean squared error. To handle this problem, the Poisson-Inverse Gaussian ridge regression estimator has been proposed as a viable alternative. This paper introduces a generalized ridge estimator to estimate regression coefficients in the Poisson-Inverse Gaussian regression model under multicollinearity. The performance of the proposed estimator is evaluated through a comprehensive simulation study, covering various scenarios and employing the mean squared error as the evaluation criterion. Furthermore, the practical applicability of the estimator is demonstrated using two real-life datasets, with its performance again assessed based on mean squared error. Theoretical analyses, supported by simulation and empirical findings, suggest that the proposed estimator outperforms existing methods, offering a more reliable solution in multicollinearity.