The Poisson regression model is widely used in statistical data modeling in diverse fields, such as epidemiology, economics, engineering, and sports. Its popularity stems from its ability to effectively describe the relationship between a statistical response variable and a set of explanatory variables. However, multicollinearity among predictors and outliers in the data can severely affect the reliability of parameter estimation, leading to inflated variances and biased results. Multicollinearity increases the sensitivity of estimators to small changes in the data, while outliers can disproportionately affect the model fit, especially in the maximum likelihood estimation framework. To address these issues, robust biased estimation techniques, such as the ridge estimator and modified ridge-type estimator, have been introduced to mitigate multicollinearity by introducing shrinkage parameters. However, the transformed M-estimator remains sensitive to outliers. In this study, we propose robust and non-robust modified Liu estimators for the Poisson regression model. The non-robust method reduces the effect of multicollinearity by developing a Liu estimator, while the proposed robust method combines transformed M-estimation with a modified form of the Liu estimator to simultaneously address multicollinearity and reduce the influence of outliers. We derive the theoretical properties of the estimators and investigate their performance through extended Monte Carlo simulations. The results indicate that the non-robust modified Liu estimator provides more stable and accurate estimates than maximum likelihood estimation, ridge estimator, and modified ridge-type estimators in the case of multicollinearity, but the robust version of the modified Liu estimator generally outperforms, especially in the presence of both multicollinearity and outliers. We also analyze a real-world dataset to demonstrate the practical value of the proposed methods.

Abstract The Zero‐inflated negative binomial (ZINB) regression models are mainly applied for count data that shows over‐dispersion and extra zeros. Multicollinearity is considered to be a significant problem in the estimation of parameters in the ZINB regression model. Thus, in order to alleviate the serious effects of multicollinearity, a new estimator is proposed which is called ZINB Stein estimator (ZINBSE). We also proposed various biasing parameters for the ZINBSE. A theoretical comparison is also conducted with some existing estimators in the literature. A Monte Carlo simulation study has been considered in order to judge the superiority of the proposed and other estimators, where the estimated mean squared error and mean absolute error are the evaluation criterion. An empirical application is also considered for illustration purposes. Based on the simulation and application results, it is observed that the new ZINBSE with proposed biasing parameters are superior over the other competitors' estimators.

Abstract This study introduces a new biased estimator called the K‐L estimator for the linear mixed model to overcome the effect of multicollinearity. We derived the mean squared error property of the proposed estimator and made a theoretical comparison with other methods. For the assessment of the K‐L estimator, we use the mean squared error criterion as a performance evaluation criterion. Moreover, we defined some shrinkage parameters for the proposed estimator. For numerical evaluation, we use a Monte Carlo simulation study and a real example. The result shows the supremacy of the K‐L estimator as compared to the available methods under certain conditions.

Summary The Poisson regression model (PRM) is usually applied in the situations when the dependent variable is in the form of count data. For estimating the unknown parameters of the PRM, maximum likelihood estimator (MLE) is commonly used. However, its performance is suspected when the regressors are multicollinear. The performance of MLE is not satisfactory in the presence of multicollinearity. To mitigate this problem, different biased estimators are discussed in the literature, that is, ridge and Liu. However, the drawback of using the traditional Liu estimator is that in most of the times, the shrinkage parameterd, attains a negative value which is the major disadvantage of traditional Liu estimator. So, to overcome this problem, we propose a new adjusted Poisson Liu estimator (APLE) for the PRM which is the robust solution to the problem of multicollinear explanatory variables. For assessment purpose, we perform a theoretical comparison with other competitive estimators. In addition, a Monte Carlo simulation study is conducted to show the superiority of the new estimator. At the end, two real life applications are also considered. From the findings of simulation study and two empirical applications, it is observed that the APLE is the most robust and consistent estimation method as compared to the MLE and other competitive estimators.

Abstract Multicollinearity poses an undesirable effect on the efficiency of the maximum likelihood estimator (MLE) in both Gaussian and non‐Gaussian regression models. The ridge and the Liu estimators have been developed as an alternative to the MLE. Both estimators possess smaller mean squared error (MSE) over the MLE. Recently, Kibria and Lukman developed KL estimator, which was found to outperform the ridge and the Liu estimators in the linear regression model. With this expectation, we developed the KL estimator for the inverse Gaussian regression model. We compare the proposed estimator's performance with some existing estimators in terms of theoretical comparison, the simulation study, and real‐life application. Smaller MSE criterion shows that the proposed estimator with one of its shrinkage parameter performs the best.

The distributions of linear combinations, and ratios of random variables arise explicitly in many reliability problems. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this paper, the exact distributions of alphaX+betaY, and X/Y are derived when X, and Y follow two of the most applied models in reliability. Computer programs for calculating the associated percentage points are also given