A comparative study of methods for transforming unbounded distributions to bounded domains: estimation and applications

This study presents four transformation methods: Inverse exponential, arctangent, rational, and reciprocal for converting unbounded probability distributions into bounded forms defined over the interval (0, 1). These methods are applied to the new XLindley distribution, a flexible model whose unbounded nature limits its applicability to enhance its utility in a bounded domain (0, 1). A comprehensive comparison assesses the effectiveness of each transformation through five important dimensions as follows: Monotonicity, mathematical properties including moments, skewness, kurtosis, and entropy, parameter estimation methods with a focus on maximum likelihood estimation and the method of Anderson-Darling, numerical simulation performance assessing convergence and stability, and practical applications as demonstrated by three real-world datasets. The results highlight each method’s weaknesses as well as strengths, with one transformation outperforming the others in terms of flexibility, estimation accuracy, and practical adaptability.