# Bivariate Kumaraswamy Distribution Based on Conditional Hazard Functions: Properties and Application

Authors: B. I. Mohammed; Md. Moyazzem Hossain; R. A. Aldallal; Mohamed S. Mohamed
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A new class of bivariate distributions is deduced by specifying its conditional hazard functions (hfs) which are Kumaraswamy distribution. The interest of this model is positively, negatively, or zero correlated. Properties and local measures of dependence of the bivariate Kumaraswamy conditional hazard (BKCH) distribution are studied. The estimation of type parameters is considered by used the maximum likelihood and pseudolikelihood of the new class. A simulation study was performed to inspect the bias and mean squared error of the maximum likelihood estimators. Finally, an application is obtained to clarify our results with the maximum likelihood and pseudolikelihood. Also, the results are used to compare BKCH distribution with bivariate exponential conditionals (BEC) and bivariate Lindley conditionals hazard (BLCH) distributions.


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## Body

## 1. Introduction

The problem of constructing new families of continuous bivariate distributions is one of the important and continue active research problem in statistics. This is due from one side to the limitations of the existing distributions and its lack in modeling some stochastic phenomena and from the other side to meet the needs of modern and developed applied field. Balakrishnon et al. [1] obtained a new technology to build the bivariate probability density functions (pdfs) with specified conditional hfs, and the development of the models determines many of the several properties, including survival function (sf), hazard bivariate function, the Clayton–Oakes measure, and the measure conditional pdfs and hfs of the marginal and conditional pdfs. Moreover, Navarro and Sarabia [2] studied the reliability properties for the series and parallel systems with component lifetimes having these dependence models.Therefore, the authors introduce a new form of the bivariate distributions with Kumaraswamy conditional hfs, and we studied some of the characteristics and discussed the estimation procedures of the parameters of this distribution. Moreover, we estimate the parameters of this distribution by two methods, e.g., maximum likelihood estimation (MLE) and maximum pseudolikelihood estimation (MPLE) procedures. Finally, this distribution is compared with other distributions considering a real dataset collected from a secondary source (p. 374 of Johnson and Wichern [3]).A continuous random variable is Kumaraswamy distribution if its pdf (Kumaraswamy [4] is(1)fXx=abxa−11−xab−1,0<x<1,a>0,b>0.The corresponding sf is(2)F¯x=1−xab,0<x<1,a>0,b>0,and the hf is(3)rx=abxa−11−xa,0<x<1,a>0,b>0.We note that(4)F¯x=exp−∫0xrudu,(5)fXx=rxexp−∫0xrudu.This distribution is applicable to some scientific, practical, and hydrological applications (Dey et al. [5]; Ishaq et al. [6]).


## 2. BKCH Model

Suppose that the conditional hfs (Kumaraswamy) are distributed as(6)r1x|y=a1b1yxa1−11−xa1,0<x,y<1,a1posativeconstant,r2y|x=a2b2xya2−11−ya2,0<x,y<1,a2posativeconstant,where b1y and b2x are function of y and x, respectively.Then, the bivariate distribution from (6) will be called a bivariate BKCH distribution.Theorem 1.
The general bivariate distribution with conditional hfs asr1x|y and r2x|y is(7)fX,Yx,y=NΘ−1xa1−1ya2−1Expθ1Log1−ya2+θ2+θ3Log1−ya2Log1−xa1,0<x,y<1,−∞<θi<∞,i=1,2,3,where NΘ=θ1,θ2,θ3 is the normalizing constant such that fX,Yx,y integrates to 1.Proof.
The conditional pdfs from (6) using (5) are(8)fX|Yx|y=a1b1yxa1−11−xa1exp−∫0xa1b1yua1−11−ua1du,(9)fY|Xy|x=a2b2xya−11−ya2exp−∫0ya2b2xua2−11−ua2du.
Then, the identityfYyfX|Yx|y=fXxfY|Xy|x yields the following relation:(10)fYya1b1y1−ya2ya2−11−xa1b1y−1=fXxa2b2x1−xa1xa1−11−ya2b2x−1,where fYy and fXx denote the marginal pdfs.
Denoting(11)gy=logfYyb1y1−ya2ya2−1,(12)hx=logfXxb2x1−xa1xa1−1, then taking logarithms in (9) and using (10) and (11) WITH then taking logarithms in (10), using (11) and (12)(13)gy+b1y−1log1−xa1−hx−b2x−1log1−ya2=0,which is a functional equation of the form ∑k=1nfkxgky=0, whose most general solution is given by Aczel [7] as(14)b1y−1=θ2+θ3log1−ya2,b2x−1=θ1+θ3log1−xa1.
From (11) and (12), we have the corresponding marginal pdfs given by(15)fXx=NΘ−1xa1−11−xa1θ2a2θ1+θ3log1−xa1+1,0<x,y<1,−∞<θi<∞,i=1,2,3,and(16)fYy=NΘ−1ya2−11−ya2θ1a1θ2+θ3log1−ya2+1,0<x,y<1,−∞<θi<∞,i=1,2,3.
Equation (7) describes the complete class of BKCH distribution that has the three parameters θ1,θ2, and θ3.
Figure1 shows the BKCH given by (7) for the special cases for any θ1, θ2, and θ3.
The conditional hfs for the BKCH distribution are(17)r1x|y=a1xa1−1θ2+θ3log1−ya2+11−xa1,0<x,y<1,−∞<θi<∞,i=1,2,3,(18)r2y|x=a2ya2−1θ1+θ3log1−xa1+11−ya2,0<x,y<1,−∞<θi<∞,i=1,2,3.
The compatibility of (17) and (18), Balakrishnan et al. [1] secures the existence of BKCH distribution. Figure 2 shows hf r1x|y of the BKCH given by (7) for the special cases for θ2 and θ3. Similarly, r2y|x.Figure 1
The pdffX,Yx,y of the BKCH distribution for selected values of θ1,θ2, and θ3. (a) θ1=1.3,θ2=.5,θ3=.2;a1=a2=0.5. (b) θ1=−1.3,θ2=−.5,θ3=−.2;a1=a2=0.5. (c) θ1=−.3,θ2=.5,θ3=0.2;a1=a2=1. (d) θ1=−.3,θ2=.5,θ3=−0.2;a1=a2=1.
(a)(b)(c)(d)Figure 2
The hfr1x|y of the BKCH distribution for selected values of θ2 and θ3. (a) θ2=.5,θ3=.2;a1=a2=0.5. (b) θ2=−.5,θ3=−.2;a1=a2=0.5. (c) θ2=.5,θ3=.2;a1=a2=1. (d) θ2=−.5,θ3=−.2;a1=a2=1.
(a)(b)(c)(d)


## 3. BKCH Properties

### 3.1. The Conditional Distributions

The conditional pdfs of the BKCH distribution are(19)fX|Yx|y=a1xa1−1θ2+θ3log1−ya2+11−xa1θ2+θ3log1−ya2,(20)fY|Xy|x=a2ya2−1θ1+θ3log1−xa1+11−ya2θ1+θ3log1−xa1,,i.e.,(21)X|Y=y∼kumaraswamyθ2+θ3log1−ya2,Y|X=x∼kumaraswamyθ1+θ3log1−xa1.Whenθ1=θ2, the conditional pdfs are identical.fX|Yx|y and fY|Xy|x given by (19) and (20) satisfy the compatibility conditions studied by Arnold et al. [8] and secure existence of BKCH distribution.


### 3.2. Regression Function

Now, using (19) and (20), the conditional kth moments are(22)EXk|Y=y=Gammaa1+k/a1Gamma2+θ2+Log1−ya2θ31+θ2+Log1−ya2θ3Gamma3+k/a1+θ2+Log1−ya2θ3,(23)EYk|X=x=Gammaa2+k/a2Gamma2+θ1+Log1−xa1θ31+θ1+Log1−xa1θ3Gamma3+k/a2+θ1+Log1−xa1θ3.Fork=1, we have(24)EX|Y=y=Gammaa1+1/a1Gamma2+θ2+Log1−ya2θ31+θ2+Log1−ya2θ3Gamma3+1/a1+θ2+Log1−ya2θ3,and(25)EY|X=x=Gammaa2+1/a2Gamma2+θ1+Log1−xa1θ31+θ1+Log1−xa1θ3Gamma3+1/a2+θ1+Log1−xa1θ3.We notice that the regression function forX|Y is nonlinear increasing for θ1,θ3<0 and decreasing for θ1,θ3>0 (Figure 3). Additionally, the regression function for X|Y is nonlinear increasing for θ2,θ3<0 and decreasing for θ2,θ3>0 (Figure 4).Figure 3
EX|Y=y of the BKCH distribution for selected values. (a) θ1=0.5,θ3=.2;a1=a2=1. (b) θ1=−0.5,θ3=−.2;a1=a2=1.
(a)(b)Figure 4
EY|X=x of the BKCH distribution for selected values of θ2 and θ3. (a) θ2=0.5,θ3=.2;a1=a2=1. (b) θ2=−0.5,θ3=−.2;a1=a2=1.
(a)(b)


### 3.3. Stochastic Ordering

Letf1x,y and f2x,y be two-bivariate BKCH distribution, and Δf12x,y=f1x,y−f2x,y such that Δf12x,y≥−Δf12y,x and Δf12x,y≥0,i.e. ,f1x,y≥f2x,y and Δf12x,y increasing in x, for all x≥y. Also, Δf12x,y decreases in y, for all y≤x. Based on the results obtained by Shanthikumar [9], then we obtain likelihood ratio order (X≥lrY, hazard rate order X≥hrY, and stochastic order X≥stY⇔Ef1x,y≥Ef2x,y, for all f1x,yandf2x,y.


## 3.1. The Conditional Distributions

The conditional pdfs of the BKCH distribution are(19)fX|Yx|y=a1xa1−1θ2+θ3log1−ya2+11−xa1θ2+θ3log1−ya2,(20)fY|Xy|x=a2ya2−1θ1+θ3log1−xa1+11−ya2θ1+θ3log1−xa1,,i.e.,(21)X|Y=y∼kumaraswamyθ2+θ3log1−ya2,Y|X=x∼kumaraswamyθ1+θ3log1−xa1.Whenθ1=θ2, the conditional pdfs are identical.fX|Yx|y and fY|Xy|x given by (19) and (20) satisfy the compatibility conditions studied by Arnold et al. [8] and secure existence of BKCH distribution.


## 3.2. Regression Function

Now, using (19) and (20), the conditional kth moments are(22)EXk|Y=y=Gammaa1+k/a1Gamma2+θ2+Log1−ya2θ31+θ2+Log1−ya2θ3Gamma3+k/a1+θ2+Log1−ya2θ3,(23)EYk|X=x=Gammaa2+k/a2Gamma2+θ1+Log1−xa1θ31+θ1+Log1−xa1θ3Gamma3+k/a2+θ1+Log1−xa1θ3.Fork=1, we have(24)EX|Y=y=Gammaa1+1/a1Gamma2+θ2+Log1−ya2θ31+θ2+Log1−ya2θ3Gamma3+1/a1+θ2+Log1−ya2θ3,and(25)EY|X=x=Gammaa2+1/a2Gamma2+θ1+Log1−xa1θ31+θ1+Log1−xa1θ3Gamma3+1/a2+θ1+Log1−xa1θ3.We notice that the regression function forX|Y is nonlinear increasing for θ1,θ3<0 and decreasing for θ1,θ3>0 (Figure 3). Additionally, the regression function for X|Y is nonlinear increasing for θ2,θ3<0 and decreasing for θ2,θ3>0 (Figure 4).Figure 3
EX|Y=y of the BKCH distribution for selected values. (a) θ1=0.5,θ3=.2;a1=a2=1. (b) θ1=−0.5,θ3=−.2;a1=a2=1.
(a)(b)Figure 4
EY|X=x of the BKCH distribution for selected values of θ2 and θ3. (a) θ2=0.5,θ3=.2;a1=a2=1. (b) θ2=−0.5,θ3=−.2;a1=a2=1.
(a)(b)


## 3.3. Stochastic Ordering

Letf1x,y and f2x,y be two-bivariate BKCH distribution, and Δf12x,y=f1x,y−f2x,y such that Δf12x,y≥−Δf12y,x and Δf12x,y≥0,i.e. ,f1x,y≥f2x,y and Δf12x,y increasing in x, for all x≥y. Also, Δf12x,y decreases in y, for all y≤x. Based on the results obtained by Shanthikumar [9], then we obtain likelihood ratio order (X≥lrY, hazard rate order X≥hrY, and stochastic order X≥stY⇔Ef1x,y≥Ef2x,y, for all f1x,yandf2x,y.


## 4. Local Measures of Dependence

Holland and Wang [10, 11] introduced a local dependence function given by the following formula:(26)γfx,y=∂2∂x∂ylnfX,Yx,y,where fX,Yx,y is the joint pdf of x and y.Theorem 2.
The BKCH distribution given by (8) is TP₂ (totally positive of order 2) iff θ3<0 and TN₂ (totally negative of order 2) iff θ3>0.Proof.
From (7), we obtain(27)γfx,y=−xa1−1ya2−11−xa11−ya2a1a2θ3.
Hence,γfx,y>0. If θ3<0, then BKCH is TP2, and γfx,y<0. If θ3>0, then BKCH is TN2.Remark 1.
(1)
It follows from theorem (2) that if the BKCH distribution, given by (7), is TP2 or TN2, then X and Y are positively or negatively, respectively, correlated.(2)
The conditional pdfs in (19) and (20) have the local measures of dependence:(28)γfx|yx,y=γfx|yx,y=γfx,y.(3)
θ3=0 iff γfx,y=0; furthermore, the two random variables X and Y are independent iff θ3=0 Balakrishnan and Lai [12].
The class has two parameters only by choosingθ3=0; the joint pdf (7) can be written as(29)fX,Yx,y=NΘ−1xa1−1ya2−1Expθ1Log1−ya2+θ2Log1−xa1,x>0,y>0,−∞<θi<∞,i=1,2,a1,a2>0.
Figure5 shows the joint pdf given by (29) for the special cases for θ1 and θ2.
The conditional hfs for the joint pdf given by (29) are(30)r1y|x=a1xa1−1−θ2+11−xa1,r2y|x=a2ya2−1−θ1+11−ya2,and the conditional pdfs are(31)fX|YX|Y=a1xa1−11−θ2exp−θ2+1log1−xa1,fY|Xy|x=a2ya2−11−θ1exp−θ1+1log1−ya2.
Now, the marginal pdfs ofX and Y using (29) are(32)fXx=NΘ−1xa1−1a21−θ1exp−θ2log1−xa1,fYy=NΘ−1ya2−1a21−θ2exp−θ1log1−ya2.
And the moment of joint pdf given by (29) is(33)EXY=NΘ−1Gamma1/aGamma1/bGamma1−θ1Gamma1−θ2a2b2Gamma2+1/b−θ1Gamma2+1/a−θ2.Figure 5
The pdffX,Yx,y given by (19) for selected values of θ1 and θ2. (a) θ1=1.3,θ2=.5;a1=a2=0.5. (b) θ1=−1.3;θ2=−.5;a1=a2=0.5. (c) θ1=−.3,θ2=.5;a1=a2=1. (d) θ1=−.3,θ2=−.5;a1=a2=1.
(a)(b)(c)(d)


## 5. Estimation of BKCH Parameters

Suppose thatxi,yi,i=1,2,…,n are observed from BKCH distribution.


### 5.1. MLE of BKCH Parameters and Asymptotic Confidence Intervals

From the BKCH distribution, the log-likelihoodlΘ function for the xi,yii=1,2,…,n sample is(34)lΘ=−nlogNΘ+a1−1∑i=1nlogxi+a2−1∑i=1nlogyi+θ1∑i=1nlog1−yia2+θ2∑i=1nlog1−xia1+θ3∑i=1nlog1−xia1log1−yia2.The likelihood equations are(35)∂NΘ/∂θ1NΘ=1n∑i=1nlog1−yia2,∂NΘ/∂θ2NΘ=1n∑i=1nlog1−xia1,∂NΘ/∂θ3NΘ=1n∑i=1nlog1−xia1log1−yia2.The MLEΘ^=θ^1,θ^2,θ^3 can be obtained numerically using systems (35). For the asymptotic confidence interval (CI), the normal approximation of the MLE can be used to construct asymptotic CIs for the parameters Θ when the sample size is large enough. A two-sided (1 − α)100% CIs for Θ are Θ^±Zα/2VarΘ^, where VarΘ^ are the asymptotic variances of Θ^.


### 5.2. MPLE of BKCH Parameters

In [13, 14], the study of alternative estimation technique is not based on normalizing constant. The MPLE for joint probability mass function (pmf) or joint pdf is based on the maximization of the product conditional pmfs or conditional pdfs, respectively. It is clear from (19) and (20). The log pseudlikelihood (logPLΘ) function is(36)logPLΘ=nloga1a2+∑i=1nlogxia1−1yia2−1+∑i=1nlogθ2+θ3log1−yia2+1+∑i=1nθ2+θ3log1−yia2log1−xia1+∑i=1nlogθ1+θ3log1−xia1+1+∑i=1nθ1+θ3log1−xia1log1−yia2.The MPLE of BKCH parameters can be obtained by solving the equations:(37)∂logPLΘ∂θ1=∑i=1n1θ1+θ3log1−xia1+1+∑i=1nlog1−yia2,∂logPLΘ∂θ2=∑i=1n1θ2+θ3log1−yia2+1+∑i=1nlog1−xia1,∂logPLΘ∂θ3=∑i=1nlog1−yia2θ2+θ3log1−yia2+1+∑i=1nlog1−xia1θ1+θ3log1−xia1+1+2∑i=1nlog1−yia2log1−xia.


## 5.1. MLE of BKCH Parameters and Asymptotic Confidence Intervals

From the BKCH distribution, the log-likelihoodlΘ function for the xi,yii=1,2,…,n sample is(34)lΘ=−nlogNΘ+a1−1∑i=1nlogxi+a2−1∑i=1nlogyi+θ1∑i=1nlog1−yia2+θ2∑i=1nlog1−xia1+θ3∑i=1nlog1−xia1log1−yia2.The likelihood equations are(35)∂NΘ/∂θ1NΘ=1n∑i=1nlog1−yia2,∂NΘ/∂θ2NΘ=1n∑i=1nlog1−xia1,∂NΘ/∂θ3NΘ=1n∑i=1nlog1−xia1log1−yia2.The MLEΘ^=θ^1,θ^2,θ^3 can be obtained numerically using systems (35). For the asymptotic confidence interval (CI), the normal approximation of the MLE can be used to construct asymptotic CIs for the parameters Θ when the sample size is large enough. A two-sided (1 − α)100% CIs for Θ are Θ^±Zα/2VarΘ^, where VarΘ^ are the asymptotic variances of Θ^.


## 5.2. MPLE of BKCH Parameters

In [13, 14], the study of alternative estimation technique is not based on normalizing constant. The MPLE for joint probability mass function (pmf) or joint pdf is based on the maximization of the product conditional pmfs or conditional pdfs, respectively. It is clear from (19) and (20). The log pseudlikelihood (logPLΘ) function is(36)logPLΘ=nloga1a2+∑i=1nlogxia1−1yia2−1+∑i=1nlogθ2+θ3log1−yia2+1+∑i=1nθ2+θ3log1−yia2log1−xia1+∑i=1nlogθ1+θ3log1−xia1+1+∑i=1nθ1+θ3log1−xia1log1−yia2.The MPLE of BKCH parameters can be obtained by solving the equations:(37)∂logPLΘ∂θ1=∑i=1n1θ1+θ3log1−xia1+1+∑i=1nlog1−yia2,∂logPLΘ∂θ2=∑i=1n1θ2+θ3log1−yia2+1+∑i=1nlog1−xia1,∂logPLΘ∂θ3=∑i=1nlog1−yia2θ2+θ3log1−yia2+1+∑i=1nlog1−xia1θ1+θ3log1−xia1+1+2∑i=1nlog1−yia2log1−xia.


## 6. Simulation Study

In this section, the MLE and MPLE approaches are used to estimate the parametersθ2, θ2, and θ3 of the BKCH distribution. The population parameters are generated using software Mathematica package. The sampling distributions are obtained for different sample sizes n = 50, 100, 200, 300, 400, and 500 from N = 500 repetitions. This study presents an assessment of the properties for both MLE and MPLE techniques in terms of bias and mean square error (MSE). A general form to generate x from one marginal fXx and then simulate a corresponding a bivariate vector using the conditional density is fY|Xy|x. The MLEs are reported in Table 1 for BKCH (1.5, 3.5, 0.05) and Table 2BKCH (1.5, 2.5, -0.05) with a1=a2=0.5.Table 1
The bias and MSE values for the BKCH (1.5, 3.5, 0.05).
MethodSizeθ1θ2θ3NBiasMSEBiasMSEBiasMSEMLE500.07920.0089−0.29010.12480.73550.48651000.07890.0085−0.28650.10790.73580.48872000.07810.00380.00740.09380.73680.43453000.07810.0028−0.29360.08790.73430.49124000.08000.0028−0.28520.07840.73410.3365500−0.08680.0025−0.29210.00040.61230.4549Table 2
The bias and MSE values for the BKCH (1.5, 2.5, -0.05).
MethodSizeθ1θ2θ3NBiasMSEBiasMSEBiasMSEMLE50−0.03590.0506−0.02061.0970.42670.1860100−0.23890.0041−0.01880.00070.42910.1851200−0.03870.0013−1.0180.00040.42760.1818300−0.23860.0011−0.01880.00030.42710.1817400−0.03970.0019−0.01730.00010.32790.1539500−0.03820.0008−0.03970.00010.43170.1069The MSEs decrease to zero asn⟶∞. This shows the consistency of the estimators.


## 7. Application of Real Data

Suppose thatX and Y are Dominant Ulna and Ulna bones. Then, the bivariate data in Table 3 represent the bone mineral density (BMD); after one year of birth, measure in gm/cm2 for 24 kids [3].Table 3
The bivariate dataset ofXi and Yi.
Children12345678Xi0.8690.6020.7650.7610.5510.7530.7080.687Yi0.9640.6890.7380.6980.6190.5150.7870.715Children910111213141516Xi0.8440.8690.6540.6920.6700.8230.7460.656Yi0.6560.7890.7260.5260.5800.7730.7290.506Children1718192021222324Xi0.6930.8830.5770.8020.5400.8040.5700.585Yi0.7400.7850.6270.7690.4980.7790,6340.640The statistic measures for the given data areμx=0.71267,σx2=0.01103,μy=0.68904,σy2=0.01242,andρx,y=0.62813. The results in Table 4 represent the MLE and MPLE of BKCH parameters.Table 4
MLE and MPLE of parameters of the BKCH.
ParametersMLEMSEMPLEMSEθ₁0.47910.0488−0.53830.0015θ₂0.42361.1580.26880.0109θ₃2.4660.93310.52830.0008The joint pdf of bivariate exponential conditionalsBPCλ1,λ2,λ3 distribution is [15](38)fX,Yx,y=expC−λ1y−λ2x+λ3xy,x>0,y>0,λ1>0,λ2>0,λ3≤0,where C is the normalizing constant.The joint pdf of bivariate Lindley conditionals hazardBLCHλ1,λ2,λ3 distribution is [16](39)fX,Yx,y=Nλ1,λ2,λ3−11+x1+yexpλ1y−λ2+λ3yx,x>0,y>0,λ1<0,λ2>0,λ3≥0,where Nλ1,λ2,λ3 is the normalized constant. BLCHλ1,λ2,λ3 is TN2; also, the conditional pdfs X|Y=y and Y|X=x are Lindley distribution with parameter λ2+λ3y and −λ1+λ3x, respectively. Table 5 includes log-likelihood and AIC and BIC of BKCH with other models.Table 5
Log-likelihood and AIC and BIC of BKCH, BLCH, and BEC distributions.
ModelParameterMLELog-likelihoodAICBICBKCHθ₁0.479151.07545.07546.308θ₂0.4236θ₃2.466BLCHλ1−0.362439.41633.41634.649λ21.0253λ31.497BECλ1−0.0631−123.799−129.796−128.563λ20.0811λ30.000496We note that AIC and BIC of the BKCH model more than the corresponding of the BLCH and BEC models which means that BKCH distribution is better to fit for the given data. The approximate 95% two-sided CI of the parametersθ1,θ2, and θ3 are given, respectively, as [−0.7548,1.7128], [−0.381,1.2282], and [1.394, 3.5379].


## 8. Conclusion

In this study, we introduced a BKCH distribution based on specified conditional hfs of Kumaraswamy distribution and its local measures of dependence. In addition, the methods of MLE and MPLE of BKHC parameters are present. In view of results, Theorem2, the interest of BKHC is positively, negatively, or zero correlated; this indicates the generality of the distribution. Furthermore, the simulation results showed that MLE operates quite uniformly, and it can be used to estimate the BKHC parameters. In particular, in Table 4, the MPLE is better than MLE because the MPLE technique uses the conditional pdfs which does not contain the normalizing constant. Therefore, Table 5 shows the BKCH distribution is a better fit for the given data compared to the BLCH and BEC distributions.



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