On a gamma–beta prime mixture model for heterogeneous positive data: properties and inference

Abstract In this paper, we introduce mixture distribution of gamma and beta prime. The gamma distribution is highly versatile for modeling continuous, positive-valued data, especially when the data exhibits skewness or variability in scale. The beta prime distribution also known as the beta distribution of the second kind, extends the beta distribution to support modeling data with heavy tails and more complex shapes. Combining these two distributions into a mixture allows the model to adapt multi modal features or heterogeneity that cannot be captured by a single distribution. This model can handle data that comes from different sources or behaves in different ways, making it more accurate and easier to understand. Some important statistical and mathematical properties of the distribution are derived such as the distribution function, moment, skewness, kurtosis, and the order statistics. The maximum likelihood estimation method is adopted to estimate the model parameters and a simulation study is conducted to investigate the efficiency of the various estimators. To evaluate the characteristics and inequality properties of the proposed distribution, we compute the Renyi and Tsallis entropy measures, Lorenz and Bonferroni curves, and the Gini index.