Ali T. Hammad

Bounded response variables, such as rates, proportions, and indices, frequently arise in the applied sciences and require regression models that respect the unit interval support, rather than relying on Gaussian regression. In this work, we propose a new regression model for continuous bounded data based on the unit Gumbel–logistic distribution as a flexible alternative to the existing models available in the literature. By reparameterizing the underlying distribution in terms of a location-type parameter directly linked to the covariate structure, our proposal enables the effect of explanatory variables on the response to be modeled in a natural and interpretable way. The proposed regression is derived from the reparameterization of the unit Gumbel–logistic distribution, which possesses a simple probability density function and a closed-form cumulative distribution function, which facilitates the derivation of its main properties. Parameter estimation is carried out using the maximum likelihood estimation method, and the finite-sample performance of the estimators is evaluated through an extensive Monte Carlo simulation study, comprising simulations for residual analysis and for the model. The proposed regression model is further compared with nine established competitors, including the beta, unit Kumaraswamy, unit Lindley, and unit log-log regression models, using two real-data applications. In both datasets, the unit Gumbel–logistic regression model achieved the highest log-likelihood and the lowest AIC, AICc, and BIC, outperforming its closest competitor, the beta regression model. Residual diagnostics and quantile-process analysis confirmed stable, well-behaved parameter estimates in both applications, supporting the unit Gumbel–logistic regression model as a valuable addition to the toolkit of regression models for bounded data.

The gamma regression model (GRM) is widely used for modeling positive, right‐skewed continuous data. The maximum‐likelihood estimator (MLE) is usually used to estimate the parameters of the GRM. However, two common issues significantly affect the reliability of MLEs: multicollinearity and outliers. Multicollinearity occurs when explanatory variables are highly correlated, which leads to difficulties in interpreting the model parameters, results that are unstable and biased, inflated variance, and an increased mean‐squared error; additionally, outliers can cause significant bias and invalid inferences and reduce the model’s efficiency. To solve these issues, we introduce a new robust estimator that can handle both multicollinearity and outliers at the same time in GRM by combining a ridge‐type estimator with effective methods like M‐estimation and Mallows‐type estimators, which results in more accurate and stable outcomes. We explain the statistical features of our new robust estimator and compare it theoretically with other estimators, as well as through Monte Carlo simulations under different conditions. The results indicate that our new estimator consistently performs better than both the standard gamma ridge estimator and other robust gamma ridge estimators when there are outliers and multicollinearity. We also applied our model to a breast cancer dataset in Wisconsin to further validate the scientific utility of the proposed estimator and support the results of the theoretical comparisons and simulations, confirming that our methodology produces more accurate coefficient estimates and better predictions compared to the existing estimators. This provides a robust framework for statisticians when multicollinearity and outliers are common problems.

Logistic regression models are widely used for analyzing binary data, with the maximum likelihood estimator (MLE) being the standard method to estimate coefficients. However, the MLE becomes unstable and unreliable in the presence of multicollinearity or outliers. Outliers distort parameter estimates by unduly influencing the likelihood function, leading to bias and poor prediction. Multicollinearity inflates the variance of coefficients, reducing stability and interpretability. While biased estimators exist for multicollinearity and robust estimators for outliers, a unified framework that simultaneously handles both issues is still lacking. To address these issues, we have proposed a class of robust ridge-type estimators that combine robust logistic estimation with shrinkage methods. A comprehensive Monte Carlo simulation study evaluated the proposed estimators under varying levels of outliers and multicollinearity. Results show that our methods consistently outperform the traditional MLE and existing estimators in terms of accuracy and robustness. Finally, we demonstrated practical utility by analyzing heavy metal and metalloid contamination levels in landfill sites in Al-Kharj, Saudi Arabia, with empirical findings confirming that the proposed robust logistic estimators provide reliable and efficient inference when both multicollinearity and outliers are present.

This study focuses on estimating the parameters of the Odd Burr XII–Exponential (OBXII‑E) distribution using both Bayesian and classical approaches. For the non‑Bayesian framework, seven estimation methods were considered: Maximum Likelihood, Least Squares, Weighted Least Squares, Maximum Product Space, Anderson–Darling, right‑Tailed Anderson–Darling, and Kolmogorov estimators. In the Bayesian context, parameter estimation was carried out using the Markov Chain Monte Carlo (MCMC) technique under different loss functions, including Squared Error, General Entropy, and Linear‑Exponential. Through extensive simulation experiments, the accuracy and consistency of each estimator were evaluated, revealing that all methods converge toward the true parameter values as sample size increases. The OBXII‑E distribution was further applied to two real datasets, where it consistently outperformed competing models based on multiple goodness‑of‑fit criteria. Overall, the results confirm the robustness and flexibility of the OBXII‑E distribution in modeling daily COVID-19 and carbon fibers data.

The Poisson inverse Gaussian regression model (PIGRM) is widely used for modeling overdispersion in count data, outperforming baseline Poisson and negative binomial regression models, particularly when dealing with longer tails and higher kurtosis data. The PIGRM coefficients are usually estimated by the maximum likelihood estimator, but its performance deteriorates in the presence of multicollinearity. Multicollinearity means high correlation between explanatory variables, resulting in numerous interactions between variables, which lead to unreliable and unstable parameter estimates, inflated variances, and increased mean squared error. To handle this issue, this study proposes an alternative biased estimator for PIGRM, named the ridge-type estimator, which combines the advantages of the Liu-type and the ridge estimators to yield more reliable and stable parameter estimates. We evaluate the proposed method’s performance through a theoretical comparison, a Monte Carlo simulation study, and an application to real-world data. Simulation results demonstrate that the proposed estimator consistently outperforms existing estimators. The proposed estimator was further validated using a real-world dataset, where multicollinearity is common. The application results confirmed the estimator’s ability to handle multicollinearity while maintaining reliable estimates. This study provides an effective estimation method for researchers working with multicollinear and overdispersed count data.

The generalized Poisson regression model (GPRM) provides a flexible framework for modeling count data, especially those exhibiting over- or underdispersion. Although the generalized Poisson maximum likelihood estimator is considered the standard method for estimating the parameters of this model, its reliability and accuracy are severely affected by the presence of multicollinearity among explanatory variables. Multicollinearity inflates the variance of parameter estimates, undermining the validity of statistical inference and ultimately leading to unstable and unreliable estimators. To mitigate these problems, this study presents the ridge estimator as a robust alternative within the GPRM framework. Several new strategies are proposed for selecting the optimal value of the ridge parameter. The statistical properties of the proposed ridge estimator were theoretically studied. Theoretical comparisons and extensive Monte Carlo simulations demonstrated a clear and significant superiority of the ridge estimator under multicollinearity conditions, confirming its robustness and efficiency. To demonstrate the scientific and practical relevance of the proposed estimator, it was applied to a real-world case study modeling carbon dioxide emissions in Canada. The results of this experimental application conclusively confirmed the simulation and theoretical comparison results, with the ridge estimator providing more stable and interpretable results than the conventional method, making it a valuable tool for researchers and decision makers in analyzing multicollinear environmental and economic data.