Asian Research Journal of Mathematics

This study conducts an in-depth exploration of the field of parametric quantile regression while introducing the new Median Based Unit Rayleigh (MBUR) distribution. The estimation process is meticulously designed through a re-parameterized maximum likelihood function, which is exemplified using a real-world dataset that effectively illustrates the underlying theoretical concepts. The author further engages in a comprehensive analysis of inference and goodness of fit. The study encompasses a thorough analysis of a real-world dataset characterized by proportions, which exposed significant deviations from the assumptions of normality and homoscedasticity. This intricate dataset revealed the presence of outliers, rendering traditional regression methods and generalized linear models ineffective for accurate analysis. Conversely, parametric quantile regression emerged as a robust alternative, adeptly addressing the challenges associated with outliers while negating the requirement for normality and accommodating heteroscedasticity. By examining the complexities inherent in proportional data, this study emphasizes the potential of both parametric quantile regression and the median-based unit Rayleigh in facilitating analysis. The core findings reveal a positive association between the employment rate and safety outcomes, whereas air pollution demonstrates a negative association in the analyses involving single predictors. However, in the context of the multivariate model, the employment rate exhibits a dominant effect. The exclusion of the employment rate results in a notable decline in likelihood, as indicated by a significant likelihood ratio test (LRT), whereas the removal of the air pollution variable yields minimal changes, accompanied by a non-significant LRT. Consequently, the recommended final model supports the inclusion of the employment rate exclusively.