On the estimation of the Wald distribution parameters with diverse applications

This paper evaluates various parameter estimation methods for the Wald distribution using simulations and real-world datasets. Simulation results confirm that increased sample sizes improve estimates, reducing bias and Root Mean-Squared Error (RMSE). The Maximum Likelihood Estimator (MLE) is generally the most robust method for large samples but unstable and biased for smaller ones, particularly in estimating λ. The Maximum product of spacing estimation (MPSE) method performs well asymptotically, with bias and RMSE decreasing as sample size increases. Least Squares (LSE) and Weighted Least Squares (WLSE) are suitable alternatives to MLE for small-to-moderate samples, showing similar estimates and reduced bias with larger samples. The Cramer-von Mises Estimator (CvME) displayed the worst efficiency due to high RMSE. Bayesian Estimators (BE) showed greater bias and lower efficiency than their frequentist counterparts, especially for λ, with performance strongly dependent on prior selection. Application to real datasets (HIV/AIDS mortality, COVID-19 death rates, and industrial gauge measurements) demonstrates the Wald distribution's feasibility in diverse health data analysis and reliability studies. The study concludes that MLE and MPSE are efficient estimation methods, suggesting the Wald distribution is a strong candidate for applied parameter estimation. Future work could focus on improving Bayesian methods via better prior selection.