MCMC Methods for Sample Generation from a New Bivariate Distribution
Llerzy Esneider Torres Ome, José Rafael Tovar Cuevas, Paula Andrea Brand Cardona · Revista Colombiana de Estadística · 2025
This article introduces a novel bivariate probability distribution derived through a transformation-based approach, along with the closed-form expression of its l-th order joint moment. Although the distribution may be employed as a prior for the shape parameters of the Beta distribution, the main focus of this work lies in evaluating the convergence behavior of Markov Chain Monte Carlo (MCMC) algorithms designed to generate samples from this new distribution. A simulation strategy is analyzed, consisting of a Gibbs sampling scheme in which an adaptive random walk Metropolis-Hastings (ARWMH) algorithm is used to sample from one of the full conditional distributions, employing a four-parameter Beta distribution as the proposal. Convergence is assessed using diagnostics such as the effective sample size (ESS) and the potential scale reduction factor (R-hat). The results show that when the elements of the parameter vector ϕ differ, the empirical moments obtained from the chains approximate the theoretical values accurately. However, when all components of ϕ are equal, the estimates of variance and covariance deviate considerably, revealing a sensitivity to symmetry in the geometry of the new distribution. A brief application in a Bayesian context is also presented, in which the new distribution is used as a prior and the Beta distribution as the likelihood. This application confirms that the proposed sampling methods yield empirical moments that are consistent with the theoretical ones, thus supporting the validity of the strategy. The contributions of this study are relevant to the design, evaluation, and implementation of MCMC techniques for sampling from complex distributions in Bayesian inference.