Bayes and Non-Bayes Estimation Methods for the Parameter of Maxwell- Boltzmann Distribution

In this paper, point estimation for parameter θ of Maxwell-Boltzmann distribution has been investigated by using simulation technique, to estimate the parameter by two sections methods; the first section includes Non-Bayesian estimation methods, such as (Maximum Likelihood estimator method, and Moment estimator method), while the second section includes standard Bayesian estimation method, using two different priors (Inverse Chi-Square and Jeffrey) such as (standard Bayes estimator, and Bayes estimator based on Jeffrey's prior). Comparisons among these methods were made by employing mean square error measure. Simulation technique for different sample sizes has been used to compare between these methods.


Introduction:
The statistical mechanic deals with Maxwell-Boltzmann distribution which describes the energy and velocity in gas, when the molecules motion freely between the level of collisions, and do not interact with each other, because of employment of the temperature of gas system. In the statistical mechanic, the Maxwell-Boltzmann distribution grants the velocity and energy in gas. In the statistical mechanic, the Maxwell-Boltzmann distribution grants the velocity and energy molecules in thermal equilibrium. In (1989) the Tyagi and Bhattacherya introduced the Maxwell distribution in lifetime model [1]. In (1998) the Chaturvedi and Rani used classical and Bayesian method of the generalized Maxwell distribution in lifetime to find the reliability estimation function [2]. In (2005) Bekker and Roux made the characteristic of reliability function to Maxwell distribution and studied Bayes estimation [3]. In (2009) Krishna and Malik estimated the reliability function to Maxwell distribution by using type-two censored sample [4].
In (2011) Kasmi, and others utilized the maximum likelihood estimator in type one censored sample of mixture Maxwell distribution [5]. In (2012) Kasmi and others utilized the Bayesian estimation for two components of Maxwell distribution by using typeone censored sample [6]. In (2013) Al-Baldawi compared between some Bayesian estimators with maximum likelihood estimator for Maxwell distribution used non-informative priors [7]. In (2016) Rasheed and Khalifa estimated the parameter of Maxwell distribution by using Bayes estimator under quadratic loss function using Noninformative prior [8]. The aim of this paper is to study four methods to find and estimate the parameter of Maxwell-Boltzmann distribution: the first is maximum likelihood method, the second is the moment method, the third is standard Bayesian method, the fourth is Jeffrey's prior method, in addition to comparing between them using Mean Square Error (MSE) utilizing Monte Carlo simulation technique with various sample sizes. Maxwell-Boltzmann Distribution [9]. Let be a random variable of Maxwell-Boltzmann distribution with one parameter, then the cumulative distribution function (cdf) of it is given: Where ( } The mean and variance for this distribution define as follows: The moment generating function (mgf) of this distribution is: 2 )] + 2 √ 2

Estimation Methods:
This study has been dealing with three estimation methods as follows:

Maximum Likelihood Estimator Method (MLEM).
The likelihood function for Maxwell-Boltzmann distribution is as follows [8]: Taking the natural logarithm for both sides, we get The partial derivative for log-likelihood function with respect to parameter , and setting the results equal to zero, we obtain The population and sample moments for parameter of Maxwell-Boltzmann distribution are given by is ME for Standard Bayes Estimator Method (SBEM) [10].
The standard Bayesian estimator method is hypothesized that unknown parameter is random variable, in some real situation, there is often additional information available about (that means there is a prior knowledge exists about the parameter ). This method is based on the notation ( ) that represents the prior distribution for parameter which came from prior knowledge, additional information, past experience. The Bayes estimator depends on the probability density function (posterior pdf) which includes information from previous knowledge and sample information [10]. Here we used two priors distributions as follows: The prior density function for where is scale parameter is: ~Invere chi-square( ) Where is a degree of freedom and = − 1 To find the posterior distribution as follows: By using squared error loss function (̂, ) = (̂− ) 2 , the risk function is: Numerical Results and Comments [12]: In this section; simulation technique is used to generate many various samples by using Monte Carlo method to compare between the methods of estimation which are mentioned in the previous section. First: generation of a sample from Maxwell distribution, we followed an algorithm suggested by Krishna and Malik (2009) [4] by the following steps: a-Generate two random numbers 1 and 2 from uniform distribution U (0, 1). b-Obtain two standard normal variates 1 and 2 using the transformation 1 = √−2 log( 1 ) 2 ( 2 ) , 2 = √−2 log( 1 ) 2 ( 2 ) , and find = 1 + 2 √2 which is (0,1). c-Repeating steps 1 and 2 three times generate a chi-square 3

Conclusions:
Throughout the estimator parameters for all four methods, we see that all the values of estimator parameters are close to the true values of parameters in Maxwell-Boltzmann distribution. Also we can see that the mean squared error procedure for all four methods have a smallest value, specially the maximum likelihood estimator method and far away from standard Bayesian estimator method.