Statistical Inference and Simulation for the Maxwell‐Boltzmann Distribution

Abstract

Statistical simulation is one approach to problem solving without experimental testing. In this paper, a method for simulating the distribution of the Maxwell‐Boltzmann distribution with MCMC approach by truncated Rayleigh distribution is presented and generated a random sample from this distribution by rejection sampling method. Some statistical inference properties for the parameter of the Maxwell‐Boltzmann distribution such as maximum likelihood estimator, method of moments estimator, uniformly minimum variance unbiased estimator and minimum risk equivariant estimator, and the relationship between maximum likelihood estimator, uniformly minimum variance unbiased estimator, and also minimum risk equivariant estimator are found. Also, the hypothesis testing is discussed and the uniform most powerful test, generalized likelihood ratio test, uniformly most powerful unbiased test and uniformly most powerful invariant test and also confidence interval with equal tails, the shortest confidence interval, unbiased confidence interval and asymptotic confidence interval for the parameter of the Maxwell‐Boltzmann model are found. By the way, a new method based on stochastic methods for finding the shortest and the unbiased confidence interval for the parameter of the Maxwell‐Boltzmann model is introduced and it is shown that with a very close approximation, it leads to the same results of previous researches that are solved by numerical methods. It is proved that the Kullback‐Leibler divergence between two Maxwell‐Boltzmann distributions with different parameters is a convex function of the ratio of the parameters and then, the Hellinger distance between these two distributions is also calculated. By selecting the multiplicative group action, the discussion of invariance is followed and maximal invariant statistics and weakly equivariant estimators are found. Next, the uniformly most powerful invariant test critical region is performed using bootstrap. In the end, using two real data series, the statistical inferences expressed in the paper are analyzed. The statistical inferences examined in this paper can also be used for the Maxwell‐Boltzmann distribution with the location parameter. Also, the unit Maxwell‐Boltzmann and the scale mixture Maxwell‐Boltzmann distributions can be generalized in the location parameter case and lead to distributions such as the truncated Maxwell‐Boltzmann distribution with the location parameter.