On Modeling X-Ray Diffraction Intensity Using Heavy-Tailed Probability Distributions: A Comparative Study

Crystallography, a cornerstone of materials science, provides critical insights into material structures through techniques such as X-ray diffraction (XRD). Among the metrics derived from XRD, intensity serves as a key parameter, reflecting the electron density distribution and offering information about atomic arrangements and sample quality. Due to its inherent variability and susceptibility to extreme values, intensity is best modeled using heavy-tailed, location-scale probability distributions. This paper investigates the model parameter estimation problem for three such distributions—log-Cauchy, half-Cauchy, and Cauchy Birnbaum–Saunders—using several methods, including maximum likelihood and the maximum product of spacings estimation methods. Monte Carlo simulations are conducted to assess the performance of these methods across various scenarios. Additionally, two real XRD intensity datasets are analyzed to compare the applicability and effectiveness of the proposed models. The results demonstrate the potential of heavy-tailed distributions for modeling XRD intensity data, providing a robust framework for future research and practical applications in material characterization.