Efficient Goodness‐of‐Fit Tests for the Maxwell–Boltzmann Distribution via Stein‐Type Characterization With Applications to (Un)censored Data

Understanding molecular motion in an ideal gas is fundamental to thermodynamics, yet directly measuring individual molecular velocities remains impractical. The Maxwell-Boltzmann (MB) distribution provides a well-established statistical model for describing the distribution of molecular speeds. In this study, we propose efficient goodness-of-fit tests for the MB distribution with unknown parameters, leveraging a novel fixed-point characterization derived from a Stein-type identity. We derive the asymptotic distribution of the proposed test statistic and establish its consistency against a broad class of alternative distributions. Furthermore, we extend the methodology to handle right-censored data, broadening its applicability to real-world scenarios where incomplete observations are common. The performance of our approach is evaluated through extensive Monte Carlo simulations, and its practical utility is demonstrated with applications to empirical data.