Open access peer-reviewed chapter

# Parameter Estimation of Weighted Maxwell-Boltzmann Distribution Using Simulated and Real Life Data Sets

Submitted: November 27th 2020Reviewed: August 31st 2021Published: October 20th 2021

DOI: 10.5772/intechopen.100227

## 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.

### Keywords

• Weighted Maxwell-Boltzmann Distribution
• prior distributions
• loss functions
• R Software

## 1. Introduction

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. This distribution is sometimes called as the distribution of velocities, energy and magnitude of momenta of molecules. Tyagi and Bhattacharya [1] who considered the Maxwell distribution as a lifetime model and discussed the Baye’s and minimum variance unbiased estimation procedures for its parameter and reliability function. Chaturvedi and Rani [2] estimated the classical and Baye’s estimators for the Maxwell distribution, after generalization it by adding another parameter. Empirical Baye’s estimation for the Maxwell distribution was also obtained by Bekker and Roux [3]. Kazmi et al. [4] derived the Bayesian estimation for two component mixture of Maxwell distribution, assuming censored data. The Maxwell-Boltzmann distribution can be used to find the distribution of particle’s kinetic energy which is related to particle’s speed by the formula E=mv2/2, provided the distribution of speed is known. The PDF of Maxwell-Boltzmann distribution is given by Maxwell [5]:

fwx=2/πθ3/2x2eθx2/2E1

And the CDF of Maxwell Distribution is given as:

Fwx=1Γα+3/2θx2/2Γα+3/2E2

Recently, Aijaz et al. [6] estimates and analyze the Bayes’ Estimators of Maxwell-Boltzmann Distribution under various Loss functions and prior Distributions. Other Contributions in Maxwell Distribution are Huang and Chen [7], Krishna and Malik [8], Tomer and Panwar [9], Zhang et al. [10], and Monisa [11].

Various Statisticians and Mathematicians have carried out the Bayesian paradigm of Maxwell-Boltzmann distribution by using loss functions and prior distributions, see Al-Baldawi [12], Dey et al. [13], Podder and Roy [14], Rasheed [15], and Spiring and Yeung [16].

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. These distributions arise, when the probability of selecting an observation varied from observation to observation. In this context, the authors generalize the Maxwell Distribution and is known as Weighted Maxwell-Boltzmann distribution. The PDF of Weighted Maxwell-Boltzmann Distribution was introduced by Aijaz et al. [19].

fwxθα=θα+3/2xα+2eθx2/22α+1/2Γα+3/2E3

Where θ is the rate parameter and ω is the weight parameter (ω>0).

Also, CDF of the Weighted Maxwell Distribution is given by:

Fwx=1Γα+3/2θx2/2Γα+3/2E4

The Reliability function and Hazard Rate of the Weighted Maxwell Distribution is given by:

Rwx=Γα+3/2θx2/2Γα+3/2E5
hwx=θα+3/2xα+2expθx2/22α+1/2Γα+3/2Γα+3/2θx2/2E6

The, rthmoments about zero of Weighted Maxwell-Boltzmann Distribution is given by:

μr=2/θr2Γα+r+3/2/Γα+3/2,Wherer=1,2,3,4,E7

In comparison to classical approach, Bayesian approach is considered to be fair enough in estimating the parameters of a distribution provided that the prior distribution describes nicely the random behavior of a parameter. Very often, priors are chosen according to one’s subjective knowledge and beliefs that is why Bayesian approach is sometimes called as subjective approach. However, Aslam [20] have shown an application of prior predictive distribution to elicit the prior density. 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.

## 2. Estimation of parameters

In this Section, the authors estimated the parameters of Weighted Maxwell-Boltzmann Distribution under Classical and Bayesian Paradigm.

### 2.1 Maximum likelihood estimation

Let x=x1x2x3xnbe a random sample of size n from Weighted Maxwell Distribution Therefore the likelihood function will be given by:

Lθω/x=θnω+32xiω+22nω+12Γω+32nexpθ2i=1nxi2E8

The, the Log likelihood function is given by:

logLθω/x=nω+32logθnω+12log2nlogΓω+32+ω+2i=1nlogxiθ2i=1nxi2E9

After, differentiating the log likelihood w.r.to θ, and equate to zero, we have:

θ̂mle=+3ni=1nxi2E10

### 2.2 Bayesian Estimation of Weighted Boltzmann Maxwell Distribution using different loss functions

#### 2.2.1 Estimation using extension of Jeffery’s prior

The Joint Probability Density Function of θand x is given by:

f1xθ=θ+3n22c1xiω+22nω+12Γω+32nexpθ2xi2E11

And, the marginal distribution function of θand x is given by:

f1x=xiω+2Γω+32nΓ+3n4c1+2222n4c1+22xi2+3n4c1+22E12

Substituting the above two Eqs. (11) and (12), we get the Posterior Probability Density function of θand x is given by:

π1θ/x=θ+3n4c12xi22+3n4c1+22Γ+3n4c1+22expθ2xi2E13

#### 2.2.1.1 Bayes’ estimator under squared error loss function

The Risk Function Under SELF is given as:

RSqEJθ̂=cθ̂2+4c+3n4c1+42+3n4c1+22xi22.4cθ̂+3n4c1+22.xi2E14

After solving the above risk function, we get the Baye’s estimator:

θ̂SqEJ=+3n4c1+2xi2E15

#### 2.2.1.2 Baye’s estimator under precautionary: Loss function

The Risk Function Under Precautionary Loss Function is given as:

RprEJθ̂=cθ̂+4c+3n4c1+42+3n4c1+22θ̂xi22.4c+3n4c1+22xi2E16

After solving the above risk function, we get the Baye’s estimator:

θ̂prEj=+3n4c1+4+3n4c1+2xi22E17

#### 2.2.1.3 Baye’s estimator under the Al-Bayyati’s loss function

The Risk function under Al-Bayyati’s Loss Function is given as:

RAlEJθ̂=2c2θ̂2Γ+3n4c1+2c2+22Γ+3n4c1+22xi2c2.+2c2+2Γ+3n4c1+2c2+62Γ+3n4c1+22xi2c2+22c2+2θ̂Γ+3n4c1+2c2+42Γ+3n4c1+22xi2c2+1E18

After solving the above risk function, we get the Baye’s estimator:

θ̂AlEJ=+3n4c1+2c2+2xi2E19

#### 2.2.1.4 Baye’s estimator under the combination of Stein’s loss function

The Risk function under the Stein’s Loss function is given as:

RStEjθ̂=θ̂xi22+3n4c12logθ̂+et1E20

After solving the above risk function, we get the Baye’s estimator:

θ̂StEj=+3n4c1xi2E21

#### 2.2.2 Bayesian estimation under gamma (α, β) prior distributions

The Joint Probability Density Function of Maxwell-Boltzmann Distribution Using Gamma Prior Distribution is given as:

f2xθ=θ+3n2xiω+22nω+12Γω̂+32nexpθ2xi2βαΓαexpβθθα1E22

Also, the Marginal density function of x is given by:

f2x=xiω+22nω+12Γω+32nβαΓαΓ+3n+2α2xi22+β+3n+2α2E23

Using above Two Results (22) and (23), we get the posterior Probability Density Function:

π2θ/x=θ+3n+2α22xi22+β+3n+2α2Γ+3n+2α2expθxi22+βE24

#### 2.2.2.1 Under squared-error loss function

The Risk function under Squared Error Function is given as:

RSqgpθ̂=cθ̂2+c+3n+2α+22+3n+2α2xi22+β22cθ̂+3n+2α2xi22+βE25

After solving the above Risk function, we get the Baye’s estimator:

θ̂Sqgp=+3n+2αxi2+2βE26

#### 2.2.2.2 Under precautionary loss function

The Risk function under precautionary Loss function is given as:

RPrgpθ̂=cθ̂+cθ̂1+3n+2α+22+3n+2α2Γ+3n+2α2Γ+3n+2α2xi22+β422c+3n+2α2xi22+β22E27

After solving the above Risk function, we get the Baye’s estimator:

θ̂Prgp=+3n+2α+2+3n+2αxi2+2βE28

#### 2.2.2.3 Under Al-Bayyati’s loss function

The Risk function under Al-Bayyati’s Loss function:

RAlgpθ̂=θ̂2Γ+3n+2α2Γ+3n+2α+2c22xi22+βc2+Γ+3n+2α+2c2+42Γ+3n+2α2xi22+βc2+22θ̂Γ+3n+2α2Γ+3n+2α+2c2+22xi22+βc2+1E29

After solving RAlgpθ̂θ̂=0for θ̂, we will have the Baye’s estimator given by:

θ̂Al,gp)=+3n+2α+2c2xi2+2βE30

#### 2.2.2.4 Under Stein’s loss function

The Risk function under Stein Loss Function is given by:

RStgpθ̂=0θ̂θlogθ̂θ1θ+3n+2α22xi22+β+3n+2α2Γ+3n+2α2expθxi22+β
RStgpθ̂=θ̂Γ+3n+2α22xi22+β+3n+2α22Γ+3n+2α22logθ̂+et1E31

After solving the above Risk function, we will have required Baye’s estimator:

θ̂Stgp=+3n+2α2xi2+2βE32

#### 2.2.3 Bayesian estimation under exponential (α) prior distributions

The Joint Probability Density Function of Weighted Maxwell-Boltzmann Distribution Using Exponential Prior Distribution is given as:

f2xθ=θ+3n2xiω+22nω+12Γω̂+32nexpθ2xi21Γαexpθθα1E33

Also, the Marginal density function of x is given by:

f2x=xiω+22nω+12Γω+32n1ΓαΓ+3n+2α2xi22+1+3n+2α2E34

Using above Two Results (33) and (34), we get the posterior Probability Density Function:

π2θ/x=θ+3n+2α22xi22+1+3n+2α2Γ+3n+2α2expθxi22+1E35

#### 2.2.3.1 Under squared-error loss function

The Risk function under Squared Error Function is given as:

RSqEpθ̂=cθ̂2+c+3n+2α+22+3n+2α2xi22+122cθ̂+3n+2α2xi22+1E36

After solving the above Risk function, we get the Baye’s estimator:

θ̂Sqep=+3n+2αxi2+2E37

#### 2.2.3.2 Under precautionary loss function

The Risk function under precautionary Loss function is given as:

RPrepθ̂=cθ̂+cθ̂1+3n+2α+22+3n+2α2Γ+3n+2α2Γ+3n+2α2xi22+1422c+3n+2α2xi22+122

After solving the above Risk function, we get the Baye’s estimator:

θ̂Prep=+3n+2α+2+3n+2αxi2+2E38

#### 2.2.3.3 Under Al-Bayyati’s loss function

The Risk function under Al-Bayyati’s Loss function:

RAlepθ̂=θ̂2Γ+3n+2α2Γ+3n+2α+2c22xi22+1c2+Γ+3n+2α+2c2+42Γ+3n+2α2xi22+1c2+22θ̂Γ+3n+2α2Γ+3n+2α+2c2+22xi22+1c2+1E39

After solving the above Risk Function, we will have the Baye’s estimator given by:

θ̂Al,Ep)=+3n+2α+2c2xi2+2E40

#### 2.2.3.4 Under Stein’s loss function

The Risk function under Stein Loss Function is given by:

RStepθ̂=0θ̂θlogθ̂θ1θ+3n+2α22xi22+1+3n+2α2Γ+3n+2α2expθxi22+1
RStepθ̂=θ̂Γ+3n+2α22xi22+1+3n+2α22Γ+3n+2α22logθ̂+et1E41

After solving the above Risk function, we will have required Baye’s estimator:

θ̂Step=+3n+2α2xi2+2E42

## 3. Simulation study of weighted Maxwell-Boltzmann distribution

In this section, we conduct the simulation studies of weighted Maxwell-Boltzmann distribution to examine the performance of the MLEs and Bayesian estimators under different prior’s like extension of Jeffrey’s’ prior, Gamma prior and Exponential prior under different loss functions in terms of expected estimates, biases, variances and mean squared errors by considering different parameter combinations. For the simulation study, sample size is taken as n = (25, 100, 300) to observe the effect of small, moderate and large samples on the estimators. We have conducted the simulation procedure for different random parameter combinations and the process was repeated 2000 times. From the simulation results, it is concluded that the performances of the Bayesian and MLEs become better when the sample size increases. In terms of MSE, the Bayesian estimators under Gamma prior perform better (see Table 1). In specific, from Table 2, extension of Jeffery’s prior under Al-Bayyati’s error loss function and stein’s loss function gives smaller MSE’s as compared to other loss functions.

nθωαβc2Criterionθ̂mlθ̂sqθpreθ̂albθste
250.50.50.50.50.5Eθ0.5047290.5278420.5262030.5115480.532156
Bias0.0047290.0278420.0262030.0115480.032156
Variance0.0037530.0071770.0060500.0039110.007792
MSE0.0037750.0079520.0067370.0040440.008826
1000.50.50.50.50.5Eθ0.5064170.5018610.5026940.5042350.500950
Bias0.0064170.0018610.0026940.0042350.000950
Variance0.0012380.0017890.0015710.0015520.001159
MSE0.0012790.0017920.0015780.0015700.001160
3000.50.50.50.50.5Eθ0.5003410.5017550.5007530.5062960.501444
Bias0.0003410.0017550.0007530.0062960.001444
Variance0.0004340.0005930.0003570.0004640.000416
MSE0.0004340.0005960.0003580.0005030.000418
251.50.50.40.50.5Eθ1.5554841.4977741.5470391.5526061.478335
Bias0.055484−0.0022260.0470390.052606−0.021665
Variance0.0590910.0551450.0579450.0540470.042449
MSE0.0621690.0551500.0601580.0568150.042918
1001.50.50.40.50.5Eθ1.5111611.4875961.4795851.5060221.510950
Bias0.011161−0.012404−0.0204150.0060220.010950
Variance0.0105590.0128940.0133870.0121100.011800
MSE0.0106830.0130480.0138040.0121460.011919
3001.50.50.40.50.5Eθ1.5001681.5106511.4930561.5147401.502537
Bias0.0001680.010651−0.0069440.0147400.002537
Variance0.0051940.0037240.0043150.0056300.004500
MSE0.0051940.0038380.0043630.0058470.004506
252.01.60.41.51.5Eθ1.9389751.9911772.0007711.9277221.956042
Bias−0.061025−0.0088230.000771−0.072278−0.043958
Variance0.0598380.0707560.1032880.0659840.061025
MSE0.0635620.0708330.1032880.0712080.062957
1002.01.60.41.51.5Eθ1.9698321.9639891.9776042.0154241.975888
Bias−0.030168−0.036011−0.0223960.015424−0.024112
Variance0.0167410.0174030.0120350.0166880.014520
MSE0.0176510.0187000.0125370.0169260.015102
3002.01.60.41.51.5Eθ2.0040781.9872081.9943962.0038042.002971
Bias0.004078−0.012792−0.0056040.0038040.002971
Variance0.0063790.0066700.0052280.0057830.008284
MSE0.0063960.0068340.0052600.0057980.008293
252.51.60.41.52.0Eθ2.3940492.4046212.3479842.5008472.458514
Bias−0.105951−0.095379−0.1520160.000847−0.041486
Variance0.0980640.1189240.1197860.1548110.094105
MSE0.1092900.1280210.1428950.1548120.095827
1002.51.60.41.52.0Eθ2.5155762.4946602.4919922.4844272.508391
Bias0.015576−0.005340−0.008008−0.0155730.008391
Variance0.0229660.0268130.0215730.0235370.023719
MSE0.0232090.0268420.0216370.0237800.023790
3002.51.60.41.52.0Eθ2.5051362.4903482.5023252.4881032.519407
Bias0.005136−0.0096520.002325−0.0118970.019407
Variance0.0087420.0082240.0060550.0105430.009009
MSE0.0087680.0083170.0060600.0106840.009386

### Table 1.

Average estimate, bias, variance and mean squared error for θ̂under gamma prior.

ml, maximum likelihood; sq, squared error loss function; pre, precautionary loss function; alb,Al-Bayyati’s loss function; ste, Stein’s loss function.

nθωc1c2Criterionθ̂mlθ̂sqθpreθ̂albθste
250.50.50.50.5Eθ0.5045500.5088410.5192920.5208390.525694
Bias0.0045500.0088410.0192920.0208390.025694
Variance0.0049790.0056600.0089700.0066780.009465
MSE0.0049990.0057380.0093420.0071120.010125
1000.50.50.50.5Eθ0.5056100.5081840.5019280.4926150.501255
Bias0.0056100.0081840.001928−0.0073850.001255
Variance0.0013130.0016770.0017070.0013180.001377
MSE0.0013440.0017440.0017100.0013720.001379
3000.50.50.50.5Eθ0.4955590.5030480.5032410.5004950.504760
Bias−0.0044410.0030480.0032410.0004950.004760
Variance0.0003910.0005000.0004670.0005320.000487
MSE0.0004110.0005100.0004780.0005330.000510
251.60.50.50.5Eθ1.6563611.6524411.5908891.5963421.624334
Bias0.0563610.052441−0.009111−0.0036580.024334
Variance0.0721860.0807210.0481460.0460700.052856
MSE0.0753630.0834710.0482290.0460840.053448
1001.60.50.50.5Eθ1.5840411.5925631.6275981.6176231.621388
Bias−0.015959−0.0074370.0275980.0176230.021388
Variance0.0137660.0137010.0122000.0113940.014179
MSE0.0140210.0137560.0129610.0117040.014636
3001.60.50.50.5Eθ1.5959421.6215901.6158411.6050641.600224
Bias−0.0040580.0215900.0158410.0050640.000224
Variance0.0061840.0064010.0056960.0039940.005462
MSE0.0062000.0068670.0059470.0040200.005462
252.51.01.50.5Eθ2.4792482.4772102.4166032.4930492.428281
Bias−0.020752−0.022790−0.083397−0.006951−0.071719
Variance0.1280380.1272450.1130670.1496210.111466
MSE0.1284680.1277650.1200220.1496700.116610
1002.51.01.50.5Eθ2.4981782.5028582.5279842.4829892.488065
Bias−0.0018220.0028580.027984−0.017011−0.011935
Variance0.0378960.0297470.0308590.0205010.023293
MSE0.0378990.0297560.0316420.0207910.023436
3002.51.01.50.5Eθ2.5105852.5006112.4903672.5074712.477393
Bias0.0105850.000611−0.0096330.007471−0.022607
Variance0.0100370.0098170.0079910.0085690.012196
MSE0.0101490.0098180.0080830.0086250.012707
252.51.00.51.5Eθ2.5664722.6909592.6070212.5591212.582644
Bias0.0664720.1909590.1070210.0591210.082644
Variance0.1336940.1325870.1328020.1336580.156276
MSE0.1381120.1690520.1442560.1371530.163106
1002.51.00.51.5Eθ2.5056122.5358152.5241662.4897362.514349
Bias0.0056120.0358150.024166−0.0102640.014349
Variance0.0426070.0298630.0299060.0317240.032759
MSE0.0426390.0311460.0304900.0318290.032965
3002.51.00.51.5Eθ2.4992792.5082992.5104902.4902292.487861
Bias−0.0007210.0082990.010490−0.009771−0.012139
Variance0.0113960.0093220.0118980.0114260.011835
MSE0.0113970.0093910.0120080.0115220.011982

### Table 2.

Average estimate, bias, variance and mean squared error for θ̂under extension of Jeffery’s prior.

ml, maximum likelihood; sq, squared error loss function; pre, precautionary loss function; alb,Al-Bayyati’s loss function; ste, Stein’s loss function.

From Table 1, we can see that the performances of the Bayesian and MLEs become better when the sample size increases. For large samples, Gamma prior under squared error loss function and Al-Bayyati’s loss function gives smaller MSE’s as compared to other loss functions and MLEs.

From Table 3, we can see that the performances of the Bayesian and MLEs become better when the sample size increases. Exponential prior under squared error loss function and stein’s loss function gives smaller MSE’s as compared to other loss functions. Thus, Exponential prior under squared error loss function and stein’s loss function can be preferred for parameter estimation.

nθωαc2Criterionθ̂mlθ̂sqθpreθ̂albθste
250.50.50.50.5Eθ0.4899540.5185420.5082900.4973230.494616
Bias−0.0100460.0185420.008290−0.002677−0.005384
Variance0.0064970.0052960.0055680.0059930.005576
MSE0.0065980.0056400.0056360.0060000.005605
1000.50.50.50.5Eθ0.5012160.5107280.4990140.5037580.508104
Bias0.0012160.010728−0.0009860.0037580.008104
Variance0.0009400.0012420.0017130.0012040.001588
MSE0.0009410.0013570.0017140.0012180.001654
3000.50.50.50.5Eθ0.5001330.4968430.4979770.4996640.499775
Bias0.000133−0.003157−0.002023−0.000336−0.000225
Variance0.0004130.0004070.0004470.0003570.000340
MSE0.0004130.0004170.0004510.0003570.000340
251.50.50.40.5Eθ1.4833791.4825641.5119031.5404511.566552
Bias−0.016621−0.0174360.0119030.0404510.066552
Variance0.0664830.0484480.0496200.0602980.047485
MSE0.0667590.0487520.0497610.0619340.051914
1001.50.50.40.5Eθ1.4904011.5085521.5012501.5016111.472398
Bias−0.0095990.0085520.0012500.001611−0.027602
Variance0.0133070.0122420.0143160.0157290.012469
MSE0.0133990.0123160.0143170.0157310.013231
3001.50.50.40.5Eθ1.5175771.5117771.5216941.4993001.505245
Bias0.0175770.0117770.021694−0.0007000.005245
Variance0.0043280.0042230.0038760.0033980.003845
MSE0.0046370.0043620.0043460.0033980.003873
252.01.60.41.5Eθ2.0449112.0096081.9856191.9808191.993491
Bias0.0449110.009608−0.014381−0.019181−0.006509
Variance0.1128420.0626200.0578350.0858260.071638
MSE0.1148590.0627120.0580420.0861940.071680
1002.01.60.41.5Eθ1.9967081.9976352.0081622.0217642.002779
Bias−0.003292−0.0023650.0081620.0217640.002779
Variance0.0152380.0127850.0134660.0147080.015945
MSE0.0152480.0127910.0135330.0151810.015953
3002.01.60.41.5Eθ1.9970462.0021541.9922722.0053032.004678
Bias−0.0029540.002154−0.0077280.0053030.004678
Variance0.0063800.0053650.0059870.0063640.006716
MSE0.0063890.0053700.0060460.0063920.006738
252.51.60.42.0Eθ2.4711142.4763972.4967942.5076172.539850
Bias−0.028886−0.023603−0.0032060.0076170.039850
Variance0.1050510.0962270.1636940.1080750.135941
MSE0.1058850.0967850.1637040.1081340.137529
1002.51.60.42.0Eθ2.5071102.4997402.4982202.5174272.508359
Bias0.007110−0.000260−0.0017800.0174270.008359
Variance0.0240000.0274710.0335770.0316660.028237
MSE0.0240510.0274710.0335810.0319700.028307
3002.51.60.42.0Eθ2.5158852.5031152.4764792.5070932.504668
Bias0.0158850.003115−0.0235210.0070930.004668
Variance0.0113820.0078550.0111950.0098510.007705
MSE0.0116350.0078650.0117490.0099010.007727

### Table 3.

Average estimate, bias, variance and mean squared error for θ̂under exponential prior.

ml, maximum likelihood; sq, squared error loss function; pre, precautionary loss function; alb,Al-Bayyati’s loss function; ste, Stein’s loss function.

## 4. Applications of weighted Maxwell-Boltzmann distribution

In this section, we present the goodness of fit of weighted Maxwell-Boltzmann distribution (WMB). For testing the goodness of fit of weighted Maxwell-Boltzmann distribution over Maxwell-Boltzmann (MB), length biased Maxwell-Boltzmann (LBMB) and area biased Maxwell-Boltzmann (ABMB) distributions, following two data sets have been considered.

Data set I is regarding tensile strength, measured in GPA, of 69 carbon fibers tested under tension at gauge lengths of 20 mm, Bader and Priest [29].

From Table 4, it has been observed that weighted Maxwell-Boltzmann distribution have the lesser AIC, AICC, −logLand BIC values as compared to Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions. Hence we can conclude that the Weighted Maxwell-Boltzmann distribution leads to a better fit than the Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions in case of analyzing the data set I.

Distributionαmlθml2loglAICBICAICC
WMB9.0791.92350.393104.787109.255104.968
MB00.47874.633151.265153.499151.325
LBMB10.63766.713135.426137.660135.485
ABMB20.79661.385124.770127.004124.829

### Table 4.

Model comparison using AIC, AICC, BIC and -logLcriterion for data set 1.

Data set II is regarding the strength data and it represents the strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows. 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 Table 5, it has been observed that weighted Maxwell-Boltzmann distribution have the lesser AIC, AICC, −logLand BIC values as compared to Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions. Hence we can conclude that the Weighted Maxwell-Boltzmann distribution leads to a better fit than the Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions in case of analyzing the data set II.

Distributionαmlθml2loglAICBICAICC
WMB9.9711.33257.656119.311123.598119.511
MB00.30881.585165.170167.313165.235
LBMB10.41174.165150.330152.473150.395
ABMB20.51369.111140.222142.366140.288

### Table 5.

Model comparison using AIC, AICC, BIC and -logLcriterion for data set II.

## 5. Conclusions

1. From the simulation Study, it was observed that the performances of the Bayesian and MLEs become better, when the sample size increases.

2. In terms of MSE, the Bayesian estimators under Gamma prior perform better. In specific, extension of Jeffery’s prior under Al-Bayyati’s error loss function and stein’s loss function gives smaller MSE’s as compared to other loss functions.

3. For large samples, Gamma prior under squared error loss function and Al-Bayyati’s loss function gives smaller MSE’s as compared to other loss functions and MLEs. Exponential prior under squared error loss function and stein’s loss function gives smaller MSE’s as compared to other loss functions.

4. Thus, Exponential prior under squared error loss function and stein’s loss function can be preferred for parameter estimation.

5. It has been observed that weighted Maxwell-Boltzmann distribution have the lesser AIC, AICC, −logLand BIC values as compared to Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions. Hence we can conclude that the Weighted Maxwell-Boltzmann distribution leads to a better fit than the Maxwell-Boltzmann, length biased Maxwell-Boltzmann and area biased Maxwell-Boltzmann distributions in case of analyzing the data set I and II.

## Acknowledgments

The authors are highly thankful to the referees and the editor for their valuable suggestions.

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© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Javaid Ahmad Reshi, Bilal Ahmad Para and Shahzad Ahmad Bhat (October 20th 2021). Parameter Estimation of Weighted Maxwell-Boltzmann Distribution Using Simulated and Real Life Data Sets, Adaptive Filtering - Recent Advances and Practical Implementation, Wenping Cao and Qian Zhang, IntechOpen, DOI: 10.5772/intechopen.100227. Available from:

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