Bayesian estimation for the Lomax distribution: comparing loss functions under different priors

Abstract

This study aims to estimate the shape parameter (𝜃) of the Lomax distribution with a known scale parameter (β) using a Bayesian approach. Three Bayesian Loss Function methods are applied: Square Error Loss Function (SELF), Entropy Loss Function (ELF), and Precautionary Loss Function (PLF). The priors used include the conjugate Gamma prior and the non-informative Jeffrey prior. The estimation was conducted on simulated data with shape parameters (𝜃 = 1.3 and 1.5) and varying sample sizes (n = 30, 150, and 300). The estimation process involves constructing the posterior distribution by combining the prior distribution with the likelihood function of the Lomax distribution. The estimated parameters are evaluated using Akaike Information Criterion (AIC), corrected AIC (AICc), and Bayesian Information Criterion (BIC) to determine the best method under various conditions. Results indicate that the Bayesian SELF method provides the best estimation with the smallest AIC, AICc, and BIC values for small sample sizes (n = 30), regardless of whether the Gamma or Jeffrey prior is used. For larger sample sizes (n = 150 and 300), the Bayesian PLF method performs better. The conjugate Gamma prior consistently produces more stable estimates compared to the non-informative Jeffrey prior. This research highlights that the optimal choice of Bayesian Loss Function depends on the sample size and the type of prior. These findings provide valuable insights for improving parameter estimation methods for the Lomax distribution, which has wide applications in heavy-tailed data analysis in fields such as finance, queuing theory, and internet traffic modeling.