A Modified Chen Inverse Rayleigh Distribution, Estimation and Application to Survival Data

Abstract

The difficulties of modeling failure rates that exhibit both monotonic and non-monotonic behavior, as well as appropriately capturing the bathtub curve in reliability engineering, remains a significant difficulty. In study, paper a Chen Inverse Raleigh (CIR) distribution within the context of the Chen-G family was developed, to eliminate the difficulty of modeling failure rates. The Probability Distribution (PDF), Cumulative Distribution Function (CDF) and the statistical properties of these CIR distributions like the survival function, hazard function, cumulative hazard function, reverse hazard function, quartile function, skewness, kurtosis, moments, linear representation and Maximum Likelihood Estimation (MLE) was also provided in the study. Our Proposed distribution provided a mild to moderate right-skewedness, while majority of the data reveals mild to moderate left-skewness, it also models leptokurtic data (heavy-tailed distributions), with kurtosis values indicating effectiveness and consistent tail behavior. A simulation study written in R provides a comparison for the CIRDs with different parameters of (β, λ, δ) and sample sizes (n) = 50, 100, 200, and 500 proved that our model converges successfully for all initial parameter settings across all sample sizes, which confirmed the robustness of the MLE optimization process. The consistency in convergence, even for smaller datasets, highlights the CIR model’s reliability and adaptability to different data conditions. The real-life data utilized to validate the efficiency of our proposed model was the survival times (in days) for patients diagnosed with head and neck cancer whose values range from 12.20 to 1776. Result showed that the CIR model demonstrates the best fit, with the lowest AIC and BIC values among the competing models.  Additionally, the CIR model achieves higher p-values in goodness-of-fit tests indicating excellent agreement between the model and the observed data.