Estimation of the Three-Parameter Inverse Rayleigh Distribution Parameters for Guinea Pig Survival Data

Abstract

The Generalized Transmuted Inverse Rayleigh Function (GTIR) distribution is an extension of the inverse Rayleigh distribution, which is commonly used to model reliability and survival data. By incorporating an additional shape parameter (α) and a transmutation parameter (λ) alongside the scale parameter (σ), this distribution offers greater flexibility in handling skewed data or data with a non-monotonic hazard function. The parameters of the GTIR distribution are estimated using the Maximum Likelihood Estimation (MLE) method; however, they must be solved implicitly through numerical procedures. In this study, the GTIR distribution was employed to analyze the survival data of guinea pigs infected with tuberculosis. The primary objective of this analysis was to estimate the distribution parameters and to provide an overview of the survival pattern. The application of the GTIR distribution to the survival and hazard functions demonstrated that guinea pigs experience a sharp decline in survival probability at the onset of tuberculosis infection, followed by a gradual decrease in the risk of mortality over time. The hazard rate pattern, which initially increases and then decreases, indicates that the most critical period occurs immediately after infection. Parameter estimation of the GTIR distribution using the MLE approach yielded estimates of λ = 0.781, α = 10.135, and σ = 12.319, confirming that this model effectively captures the complex survival pattern with high accuracy.