Abstract
This paper deals with estimation of parameters of Weighted Maxwell-Boltzmann Distribution by using Classical and Bayesian Paradigm. Under Classical Approach, we have estimated the rate parameter using Maximum likelihood Estimator. In Bayesian Paradigm, we have primarily studied the Bayes’ estimator of the parameter of the Weighted Maxwell-Boltzmann Distribution under the extended Jeffrey’s prior, Gamma and exponential prior distributions assuming different loss functions. The extended Jeffrey’s prior gives the opportunity of covering wide spectrum of priors to get Bayes’ estimates of the parameter – particular cases of which are Jeffrey’s prior and Hartigan’s prior. A comparative study has been done between the MLE and the estimates of different loss functions (SELF and Al-Bayyati’s, Stein and Precautionary new loss function). From the results, we observe that in most cases, Bayesian Estimator under New Loss function (Al-Bayyati’s Loss function) has the smallest Mean Squared Error values for both prior’s i.e., Jeffrey’s and an extension of Jeffrey’s prior information. Moreover, when the sample size increases, the MSE decreases quite significantly. These estimators are then compared in terms of mean square error (MSE) which is computed by using the programming language R. Also, two types of real life data sets are considered for making the model comparison between special cases of Weighted Maxwell-Boltzmann Distribution in terms of fitting.
Highlights
Kazmi et al [4] derived the Bayesian estimation for two component mixture of Maxwell distribution, assuming censored data
Single fibers were tested under tension at gauge lengths of 10 mm with sample sizes n = 63; see Bader and Priest [29] and Surles and Padgett [30]
From the simulation Study, it was observed that the performances of the Bayesian and MLEs become better, when the sample size increases
Summary
In Statistical Mechanics, there are a lot of applications of Maxwell-Boltzmann Distribution. The Maxwell-Boltzmann distribution forms the basis of the kinetic energy of gases, which explains many fundamental properties of gases, including pressure and diffusion. Aijaz et al [6] estimates and analyze the Bayes’ Estimators of Maxwell-Boltzmann Distribution under various Loss functions and prior Distributions. The concept of weighted distributions introduced by Fisher [17] and later it was formulated in general terms by Rao [18] in connection with modeling statistical data. These Distributions are applicable, when each and every observation is given an equal chance of being recorded. The Reliability function and Hazard Rate of the Weighted Maxwell Distribution is given by: RwðxÞ 1⁄4 Γððα þ 3Þ=2Þ (5). A number of symmetric and asymmetric loss functions have been shown to be functional, see Kasair et al [21], Norstrom [22], Reshi et al [23], Zellner [24], Reshi et al [25], Dey and Maiti [26], Alkutbi [27], Wald [28], etc
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