An exploration of the Bose–Einstein distribution

Modeling of physical phenomena has been introduced by many physicists. In this paper, we explore some particle models with the Bose–Einstein distribution. The statistical inference of these distributions including moments, maximum likelihood estimation, Fisher information, entropy, and Kullback–Leibler divergence is analyzed. Given the maximum likelihood estimator and Fisher information estimate, we can use the Wald test to build confidence intervals and test hypotheses. Random sample generation based on the Metropolis–Hastings algorithm and the integral probability transformation theorem is investigated. We prefer to use the integral probability transformation theorem because of its speed and high accuracy. Also, for two independent random variables from the Bose–Einstein distributions, the distribution of their functions, as well as their moments, is examined. To evaluate the goodness-of-fit of the proposed models, we analyze real-world salary data across different levels. Graphical tools including histograms, empirical density functions, empirical cumulative distribution functions, and Q–Q plots are employed to visually compare the data with the fitted distributions. Bootstrap and permutation methods are used to compute the Kolmogorov–Smirnov statistic and the Kullback–Leibler divergence, while Wald, likelihood ratio, and score tests are applied for hypothesis testing. Comparisons with other common distributions (exponential, Weibull, log-normal, and gamma) are also conducted. The results consistently indicate that the Bose–Einstein type I distribution provides a substantially better fit to the empirical data compared to the Bose–Einstein type II distribution and the other models considered, as confirmed by residual analysis, descriptive measures, and formal test statistics. Finally, modeling with real data is carried out. In the conclusion section, suggestions are made for future research that can be conducted as a continuation of this study. Also, the R codes are reported in Appendix A.