GIBBS SAMPLING FOR AN EXPONENTIA TED POWER LOMAX DISTRIBUTION WITH DIFFERENT PRIORS

Abstract

Bayesian inference, particularly Gibbs sampling, is a powerful tool for parameter estimation in high-dimensional distributions, effectively handling complex integrations. In this study, we employ Gibbs sampling to estimate the four parameters (α, β, γ and c) of the Exponentiated Power Lomax (EPOLO) distribution, a flexible model widely applicable in reliability analysis and survival studies. To ensure accurate parameter estimation, we assess the model's performance using three prior distributions: Uniform, Exponential, and Gamma. The reliability of the estimates is evaluated through bias and mean squared error (MSE) computations, while the convergence of the Gibbs sampling process is verified using trace plots and histograms, confirming stationarity and the absence of autocorrelation. The model fit is further assessed using multiple goodness-of-fit criteria, including:
• Information-theoretic measures: Akaike Information Criterion (AIC), Corrected AIC (CAIC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC).
• Distributional fit tests: Kolmogorov-Smirnov Statistic (KSS) and p-values to evaluate agreement between the model and observed data.
• Posterior predictive checks, comparing posterior means with observed data, to validate predictive accuracy.
Our findings reveal that the Uniform prior provides the most accurate and stable parameter estimates, with lower bias, reduced MSE, and superior model fit. In contrast, the Exponential prior exhibits a tendency to overestimate predictions, while the Gamma prior results in numerical instability and extreme parameter overestimation. These results highlight the effectiveness of Bayesian estimation for the EPOLO distribution and emphasize the importance of prior selection in achieving robust parameter estimates.