Bilal-G Family of Distributions with Applications to Biomedical and Reliability Engineering Data

Abstract

This paper introduces the Bilal-G (B-G) family of distributions, a novel generator-based method for enhancing the flexibility of existing probability models to better accommodate complex data structures prevalent in biomedical and reliability engineering. Data from these fields frequently exhibit features like high skewness, significant outliers, and non-monotone hazard rates that challenge conventional distributions. Using the Bilal distribution as the generator, we construct the new family’s general cumulative distribution function (CDF) and probability density function (PDF), from which a key, parsimonious sub-model, the two-parameter Bilal-Exponential (BE) distribution, is derived. We thoroughly analyze the BE distribution’s properties, including its capability to model an increasing hazard rate, which is supported by Total Time on Test (TTT) plots of the application datasets. A comprehensive simulation study evaluates the performance of fifteen distinct non-Bayesian estimators, revealing that the Minimum Spacing Linex Distance (MSLNDE) method consistently provides the most accurate and precise parameter estimates across various sample sizes. Finally, the superiority of the BE distribution is demonstrated through its successful application to two real datasets: one on patient mortality rates and one on component failure times. For the mortality data (Data I), the BE model reduced the Akaike Information Criterion (AIC) by 1.99 units compared to the classical Weibull distribution. For the component failure data (Data II), the Bayesian Information Criterion (BIC) was reduced by 0.41 units compared to the best-fitting competing model (TIHTE), confirming the BE distribution’s exceptional goodness-of-fit and reliability as a practical lifetime model.