Analysis and Simulation of Lucas Distribution Using Markov Chain Monte Carlo Methods

This study aims to investigate the properties of the Lucas distribution and apply the Markov Chain Monte Carlo (MCMC) method to simulate it and estimate its statistical characteristics. The work begins by defining the probability mass function of the Lucas distribution and establishing its mathematical connection to Birth–Death processes in continuous-time Markov chains. A generator matrix is constructed, and the corresponding discrete-time transition matrix is derived, demonstrating that the stationary distribution matches the target Lucas distribution. The Metropolis–Hasting’s algorithm is then implemented in the R programming environment to generate samples from this distribution. The simulated results are analyzed and compared with theoretical values through graphical and statistical summaries. The findings reveal a high degree of agreement between the estimated and theoretical values over most of the range, with noticeable underrepresentation in the upper tail, suggesting the need for improved proposal mechanisms or longer chains. This research provides both a mathematical framework and an applied methodology for using MCMC to simulate uncommon discrete distributions and offers methodological enhancements to overcome the observed limitations.