Abstract
In this paper, Bayes estimators of the parameter of Maxwell distribution have been derived along with maximum likelihood estimator. The non-informative priors; Jeffreys and the extension of Jeffreys prior information has been considered under two different loss functions, the squared error loss function and the modified squared error loss function for comparison purpose. A simulation study has been developed in order to gain an insight into the performance on small, moderate and large samples. The performance of these estimators has been explored numerically under different conditions. The efficiency for the estimators was compared according to the mean square error MSE. The results of comparison by MSE show that the efficiency of Bayes estimators of the shape parameter of the Maxwell distribution decreases with the increase of Jeffreys prior constants. The results also show that values of Bayes estimators are almost close to the maximum likelihood estimator when the Jeffreys prior constants are small, yet they are identical in some certain cases. Comparison with respect to loss functions show that Bayes estimators under the modified squared error loss function has greater MSE than the squared error loss function especially with the increase of r.
Highlights
In physics statistical mechanics, the Maxwell–Boltzmann distribution describes particle speeds in gases, where the particles move freely between short collisions, but do not interact with each other, as a function of the temperature of the system, the mass of the particle, and speed of the particle[1]
The Maxwell distribution gives the distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics[2].The Maxwell distribution was first introduced in the literature as a lifetime model by Tyagi and Bhattacharya (1989)[3]
In general, comparison shows that Bayes' estimator of the parameter θ of Maxwell distribution based on Jeffreys prior with respect to the squared error loss function gives less mean square errors (MSE) than the extension of Jeffreys prior, only when θ is small
Summary
In physics statistical mechanics, the Maxwell–Boltzmann distribution describes particle speeds in gases, where the particles move freely between short collisions, but do not interact with each other, as a function of the temperature of the system, the mass of the particle, and speed of the particle[1]. The Maxwell distribution gives the distribution of speeds of molecules in thermal equilibrium as given by statistical mechanics[2].The Maxwell distribution was first introduced in the literature as a lifetime model by Tyagi and Bhattacharya (1989)[3] They obtained Bayes estimates and minimum variance unbiased estimators of the parameter and reliability function. The obtained Baysian estimates of the shape parameter θ are compared to its maximum likelihood counterpart The performance of these estimates is assessed using Monte Carlo simulation study, considering various sample sizes; several specific values of the parameter θ and Jeffreys prior constants. Bayes estimators for the parameter θ, was considered with non-informative priors. Following is the derivation of these estimators i )Jeffreys prior information, under squared error loss function
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