A New Modified Arctan Model: Statistical Properties, Estimation Methods, Simulation Study, and Real-Life Application

Abstract

This study introduces the modified Weibull Arctan (MWArctan) distribution, a new model derived from the Modified Weibull-G (MWG) family using the Arctan function as a generator. The theoretical formulation of the distribution is presented through derivations of its probability density function, cumulative distribution function, survival function, and hazard function. Analytical expansions are developed to simplify the computation of statistical properties, moments, and the quantile function. Three parameter estimation methods were evaluated through a Monte Carlo simulation across various sample sizes. The MLE method consistently produced the most efficient and stable estimates, exhibiting the lowest bias, MSE, and RMSE values. In contrast, LSE demonstrated lower bias only for small samples, but higher variance. The practical applicability of the MWArctan model was verified using real-life data from 40 turbochargers in diesel engines. The proposed model achieved the best fit compared to six competing Arctan-based distributions, as indicated by specific information criteria and goodness-of-fit tests. Graphical comparisons confirmed its superior flexibility in capturing both the central tendency and tail behavior of the empirical data. These results establish MWArctan as a robust and adaptable model for analyzing reliability and lifetime data.