Comparative Study of the Median Based Unit Rayleigh and its Generalized Form the Generalized Odd Median Based Unit Rayleigh

1.1. MBUR and GOMBUR

In previous work iman Attia (Iman M. Attia, 2024) discussed the MBUR. Y is a random variable distributed as MBUR with the following PDF, CDF respectively as shown in equation 1 and 2

$$f\left(y\right)={\displaystyle \frac{6}{{\alpha }^2}}\left[1-y^{{\displaystyle \frac{1}{{\alpha }^2}}}\right]y^{\left({\displaystyle \frac{2}{{\alpha }^2}}-1\right)},\hspace{1em}\hspace{1em}0<y<1,\hspace{1em}\hspace{1em}\alpha >0$$

(1)

$$F\left(y\right)={3y}^{{\displaystyle \frac{2}{{\alpha }^2}}}-{2y}^{{\displaystyle \frac{3}{{\alpha }^2}}},\hspace{1em}\hspace{1em}0<y<1,\hspace{1em}\hspace{1em}\alpha >0$$

(2)

In previous work iman Attia (Iman M. Attia, 2025) discussed the GOMBUR. Y is a random variable distributed as GOMBUR version 1 and version 2 with the following PDFs respectively as shown in equation 3 and 4

$$f_y\left(y\right)={\displaystyle \frac{\mathsf{\Gamma }\left(2n+2\right)}{\mathsf{\Gamma }\left(n+1\right)\mathsf{\Gamma }\left(n+1\right)}}{\displaystyle \frac{1}{{\alpha }^2}}{\left[1-y^{{\alpha }^{-2}}\right]}^n{\left[y\right]}^{{\displaystyle \frac{n+1}{{\alpha }^2}}-1},n\ge 0,\alpha >0,0<y<1$$

(3)

$$f_y\left(y\right)={\displaystyle \frac{\mathsf{\Gamma }\left(n+1\right)}{\mathsf{\Gamma }\left(\frac{n}{2}+\frac{1}{2}\right)\mathsf{\Gamma }\left(\frac{n}{2}+\frac{1}{2}\right)}}{\displaystyle \frac{1}{{\alpha }^2}}{\left[1-y^{{\alpha }^{-2}}\right]}^{{\displaystyle \frac{n-1}{2}}}{\left[y\right]}^{{\displaystyle \frac{n+1}{2{\alpha }^2}}-1},n\ge 1,\alpha >0,0<y<1$$

(4)

And the following CDFs for version 1 and version 2 respectively as shown in equation 5

$$P\left(Y<y\right)=I_w\left(n+1,n+1\right) \mathrm{for} \mathrm{version} 1 \text{\&} P\left(Y<y\right)=I_w\left({\displaystyle \frac{n+1}{2}},{\displaystyle \frac{n+1}{2}}\right) \mathrm{for} \mathrm{version} 2$$

(5)

1.2. Maximum Likelihood Estimation

Let Y is a random variable having the PDF of GOMBUR-1. To derive the MLE for version 1, for one observation taking the log of equation 3 results into equation 6:

$$l\left(y;\alpha ,n\right)=\ln \mathsf{\Gamma }\left(2n+2\right)-\ln \mathsf{\Gamma }\left(n+1\right)-\ln \mathsf{\Gamma }\left(n+1\right)+\ln {\alpha }^{-2}+n\ln \left[1-y_i^{{\alpha }^{-2}}\right]+\left({\displaystyle \frac{n+1}{{\alpha }^2}}-1\right)ln{y}_i$$

(6)

Take the first derivative of equation 6 with respect to n and alpha parameter yields equation 7 and 8 resectively:

$${\displaystyle \frac{\partial l}{\partial n}}=2\ast \psi \left(2n+2\right)-\psi \left(n+1\right)-\psi \left(n+1\right)+\ln \left[1-y_i^{{\alpha }^{-2}}\right]+{\alpha }^{-2}\ln y_i$$

(7)

$${\displaystyle \frac{\partial l}{\partial n}}={\displaystyle \frac{-2}{\alpha }}+\left({\displaystyle \frac{2n}{{\alpha }^3}}\right){\displaystyle \frac{y_i^{{\alpha }^{-2}}\ln y_i}{1-y_i^{{\alpha }^{-2}}}}-\left({\displaystyle \frac{2\left[n+1\right]}{{\alpha }^3}}\right)\ln y_i$$

(8)

Let Y is a random variable having the PDF of GOMBUR-2. To derive the MLE for version 2, for one observation taking the log of equation 4 results into equation 9:

$$l\left(y;\alpha ,n\right)=\ln \mathsf{\Gamma }\left(n+1\right)-\ln \mathsf{\Gamma }\left({\displaystyle \frac{n+1}{2}}\right)-\ln \mathsf{\Gamma }\left({\displaystyle \frac{n+1}{2}}\right)+\ln {\alpha }^{-2}+\left({\displaystyle \frac{n-1}{2}}\right)\ln \left[1-y_i^{{\alpha }^{-2}}\right]+\left({\displaystyle \frac{n+1}{2{\alpha }^2}}-1\right)ln{y}_i$$

(9)

Take the first derivative of equation 9 with respect to n and alpha parameter yields equation 10 and 11 respectively:

$${\displaystyle \frac{\partial l}{\partial n}}=\psi \left(n+1\right)-{\displaystyle \frac{1}{2}}\psi \left({\displaystyle \frac{n+1}{2}}\right)-{\displaystyle \frac{1}{2}}\psi \left({\displaystyle \frac{n+1}{2}}\right)+{\displaystyle \frac{1}{2}}\ln \left[1-y_i^{{\alpha }^{-2}}\right]+{\displaystyle \frac{1}{2}}{\alpha }^{-2}\ln y_i$$

(10)

$${\displaystyle \frac{\partial l}{\partial n}}={\displaystyle \frac{-2}{\alpha }}+\left({\displaystyle \frac{n-1}{{\alpha }^3}}\right){\displaystyle \frac{y_i^{{\alpha }^{-2}}\ln y_i}{1-y_i^{{\alpha }^{-2}}}}-\left({\displaystyle \frac{n+1}{{\alpha }^3}}\right)\ln y_i$$

(11)

For each version set the above equations to zero and since they are non-linear equations, numerical methods like quasi-Newton method can be used as a solution.

2. Description of the Real Data:

The real data used in this paper can be found in the Appendix A at the end of the paper; see Table A1. These are 14 datasets. The first five datasets were used by the author (Iman M. Attia, 2024) in previous work. The first dataset is the dwelling without basic facilities. The second one is the quality support network. The third one is the educational attainment dataset. The fourth one is the flood data. The fifth one is the time between failures dataset. The sixth dataset is the COVID-19 death rate in Canada previously discussed by (Nasiru et al., 2022) . The seventh dataset is the COVID-19 death rate in Spain previously discussed by (Ahmed Z. Afify et al., 2022) . The eighth dataset is the COVID-19 death rate in United Kingdom previously discussed by (Nasiru et al., 2022) . The ninth dataset is the 48 rock samples from a petroleum reservoir previously discussed by (Gauss Moutinho Cordeiro & Rejane dos Santos Brito, 2010) . The tenth dataset is the daily snowfall amounts of 30 observations measured in inches of water taken from non-seeded experimental units, which was conducted in the vicinity of Climax Colorado previously discussed by (Ishaq et al., 2023). The eleventh dataset is about the total milk production in the first birth 107 cows from SINDI race. These cows are property of the Carnauba farm located in Brazil previously mentioned by (Gauss Moutinho Cordeiro & Rejane dos Santos Brito, 2010). The twelfth dataset is the COVID-19 recovery rate in Spain previously mentioned by (Ahmed Z. Afify et al., 2022). The thirteen dataset is the voter turnout dataset previously mentioned by (Iman M.Attia, 2024) . The fourteen dataset is the unit capacity factors previously mentioned by (Maya et al., 2024) .

3. Real Data Analysis

The analysis of the data sets points to conclude how these sets support the following distributions: Beta, Topp Leone, Unit Lindely, Kumaraswamy. The fitting of these data sets will be compared to the fitting of the new MBUR distribution and its generalized forms GOMBUR-1 and GOMBUR-2. The metrics used for this evaluation include the following: LL (log-likelihood), Akaike Information Criterion (AIC), corrected AIC (CAIC), Bayesian Information Criterion (BIC), and Hannan-Quinn Information Criterion (HQIC). Additionally, the Kolmogorov-Smirnov (K-S) test will be conducted. The test's results will include its value, along with the outcome of the null hypothesis (H0), which proposes that the data set follows the investigated distribution; if this assumption is not encountered, the null hypothesis will be rejected. The P-value for the test will also be documented. Furthermore, the Cramér-von Mises test and the Anderson-Darling test will be implemented, with their respective values stated. Figures depicting the empirical cumulative distribution function (eCDF) and the theoretical cumulative distribution functions (CDF) of the distributions will be illustrated, each in its place. Finally, the values of the estimated parameters, along with their estimated variances and standard errors, will be reported. The competitors’ distributions are:

1- Beta Distribution:

$$f\left(y;\alpha ,\beta \right)={\displaystyle \frac{\mathsf{\Gamma }\left(\alpha +\beta \right)}{\mathsf{\Gamma }\left(\alpha \right)\mathsf{\Gamma }\left(\beta \right)}}{y}^{\alpha -1}{\left(1-y\right)}^{\beta -1},0<y<1,\alpha >0,\beta >0$$

2- Kumaraswamy Distribution:

$$f\left(y;\alpha ,\beta \right)=\alpha \beta y^{\alpha -1}{\left(1-y^{\alpha }\right)}^{\beta -1},0<y<1,\hspace{1em}\hspace{1em}\alpha >0,\beta >0$$

3- Median Based Unit Rayleigh:

$$f\left(y;\alpha \right)={\displaystyle \frac{6}{{\alpha }^2}}\left[1-y^{{\displaystyle \frac{1}{{\alpha }^2}}}\right]y^{\left({\displaystyle \frac{2}{{\alpha }^2}}-1\right)},0<y<1,\hspace{1em}\hspace{1em}\alpha >0$$

4- Topp-Leone Distribution:

$$f\left(y;\theta \right)=\theta \left(2-2y\right){\left(2y-y^2\right)}^{\theta -1},0<y<1,\hspace{1em}\hspace{1em}\theta >0$$

5- Unit-Lindley:

$$f\left(y;\theta \right)={\displaystyle \frac{{\theta }^2}{1+\theta }}{\left(1-y\right)}^3\exp \left({\displaystyle \frac{-\theta y}{1-y}}\right),\hspace{1em}\hspace{1em}0<y<1,\hspace{1em}\hspace{1em}\theta >0$$

Comparison tools are: (k) is the number of parameter, (n) is the number of observations.

$$AIC=-2MLL+2k,\hspace{1em}\hspace{1em}CAIC=-2MLL+{\displaystyle \frac{2k}{n-k-1}},\hspace{1em}\hspace{1em}BIC=-2MLL+k\log \left(n\right)$$

$$HQIC=-2\log L+2k\ast \ln \left[\ln \left(n\right)\right]$$

$$KS-test={Sup}_n\left|F_n-F\right|,\hspace{1em}{F}_n={\displaystyle \frac{1}{n}}\sum _{i=1}^n{I}_{x_i<x}$$

$$Cramer-Von-Mises-test(CVM)={\displaystyle \frac{1}{12n}}+\sum _{i=1}^n{\left\{F\left(x_i\right)-{\displaystyle \frac{2i-1}{2n}}\right\}}^2$$

$$Anderson-Darling-test(AD)=-n-{\sum }_{i=1}^n\left({\displaystyle \frac{2i-1}{n}}\right)\left\{log\left[F\left(x_i\right)\right]+log\left[1-F\left(x_{n-i+1}\right)\right]\right\}$$

3.1. Descriptive statistics

Figure 1, Figure 2 and Figure 3 illustrates the boxplot of the data sets

Table 1 and Figure 1, Figure 2 and Figure 3 illustrate different shapes and characteristics of data.

3.2. Individual Dataset Analysis (Results & Discussion)

For each datasets a table showing the results of the fitted distribution and 6 figures highlight the fitted CDFs and PDFs for the tested distributions, the PP plots of these distributions and the MBUR, GOMBUR-1& 2 versions. There is an additional figure to compare the fitted CDFs and the PDFs of the two versions. Table 2 discusses the results of the dwelling without basic facilities datasets.

Fitting GOMBUR-1 & GOMBUR-2 introduces a reduction in the covariance between the parameters alpha and n which indicates less association between the 2 parameters. This is in comparison with the high association seen between the 2 parameters of the Beta and Kumaraswamy distributions relative to that seen in the GOMBUR1 & GOMBUR-2.

Table 2. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=5.7248$		$n=12.4496$
	$\alpha =2.4988$		$\alpha =2.4988$
Var	2.7974	0.0255	11.1896	0.0509
	0.0255	0.008	0.0509	0.008
SE(n)	0.3004		0.6008
SE(a)	0.0161		0.0161
AIC	-158.1462		-158.1462
CAIC	-157.7176		-157.7176
BIC	-155.2782		-155.2782
HQIC	-157.2113		-157.2113
LL	81.0731		81.0731
K-S Value	0.1654		0.1654
H0	Fail to reject		Fail to reject
P-value	0.3279		0.3279
AD	0.8727		0.8727
CVM	0.1515		0.1515
Determinant	0.0218		0.0874
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 2. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=5.7248$		$n=12.4496$
	$\alpha =2.4988$		$\alpha =2.4988$
Var	2.7974	0.0255	11.1896	0.0509
	0.0255	0.008	0.0509	0.008
SE(n)	0.3004		0.6008
SE(a)	0.0161		0.0161
AIC	-158.1462		-158.1462
CAIC	-157.7176		-157.7176
BIC	-155.2782		-155.2782
HQIC	-157.2113		-157.2113
LL	81.0731		81.0731
K-S Value	0.1654		0.1654
H0	Fail to reject		Fail to reject
P-value	0.3279		0.3279
AD	0.8727		0.8727
CVM	0.1515		0.1515
Determinant	0.0218		0.0874
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 2 shows marked improvement in the indices for the fitted GOMBUR-1 & GOMBUR-2 with increased negativity of AIC, CAIC, BIC and HQIC to levels more than that of beta, Kumaraswamy and MBUR. Log-likelihood increased to levels above that of the beta, kumaraswamy and MBUR. Tests statistics of KS, AD and CVM show lowered values than those of beta, kumaraswamy and MBUR. The determinants of both GOMBUR-1 & GOMBUR-2 are less than that of the Beta distribution but higher than that of the Kumaraswamy distribution. The overall assessment of generalization shows that both GOMBUR-1 & GOMBUR-2 outperform the beta, Kumaraswamy and MBUR distribution. Figure 1 shows how the fitted GOMBUR-1 CDF aligns better with the fitted CDFs of both Kumaraswamy and Beta distributions than the fitted CDF of BMUR. Figure 2 also shows how the fitted GOMBUR-1 PDF aligns better than with the fitted PDFs of both Beta and Kuamraswamy distributions than the fitted PDF of the MBUR. The determinant of GOMBUR-1 is less than that of the GOMBUR-2 which points to the fact that the GOMBUR-1 is more efficient than GOMBUR-2 in fitting the data for the same level of other indices like AIC & BIC and tests like KS, AD & CVM. GOMBUR-1 has higher AIC & BIC and log-likelihood than that of the Kumaraswamy, although Kumaraswamy has less determinant than that of GOMBUR-1. It is a tradeoff between the biasness and efficiency between the 2 distributions.

Figure 3 shows the PP plot of the fitted Beta distribution and the QQ plot for the fitted Kumaraswamy. Figure 4 and Figure 5 show PP plot comparison between the fitted MBUR, GOMBUR-1 & GOMBUR-2 distributions. Figure 6 shows a comparison between the theoretical CDF and PDF of the fitted GOMBUR 1 & 2 for the dwelling dataset. The fitted CDFs and PDFs of both versions are identical and this is reflected on the indices obtained by estimation. Table 3 illustrates the Quality support network analysis results. Table 3 shows the results of quality network analysis.

Table 4 shows that the 5 distributions fit the data and the MBUR outperform all of them followed by the Unit Lindley. Figure 7 shows more or less perfect alignment of the theoretical CDFs of Beta, Kumaraswamy and GOMBUR-1 distributions, although all the fitted distributions align well with each other. Figure 8 shows the perfect alignment of the graphs of the PDFs of the three distributions: the Beta, Kumaraswamy and GOMBUR-1 distributions, this can be a reflection of having nearly equaled AIC, CAIC, BIC & HQIC. Figure 9, Figure 10 and Figure 11 shows varieties of graphs of PP plots and QQ plots for the competitor distributions. Figure 12 illustrates the comparison between the fitted CDFs and fitted PDFs of both GOMBUR-1 and 2.

Table 3. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.4765$		$n=3.9529$
	$\alpha =0.3651$		$\alpha =0.3651$
Var	0.5554	0.0056	2.2216	0.0113
	0.0056	0.00075961	0.0113	0.00075961
SE(n)	0.1666		0.3333
SE(a)	0.0062		0.0062
AIC	-56.5514		-56.5514
CAIC	-55.8455		-55.8455
BIC	-54.5599		-54.5599
HQIC	-56.1626		-56.1626
LL	30.2757		30.2757
K-S Value	0.0977		0.0977
H0	Fail to reject		Fail to reject
P-value	0.9458		0.9458
AD	0.3728		0.3728
CVM	0.0545		0.0545
Determinant	0.00039		0.0016
Significant(n)	0		0
Significant(a)	0		0

Table 3. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.4765$		$n=3.9529$
	$\alpha =0.3651$		$\alpha =0.3651$
Var	0.5554	0.0056	2.2216	0.0113
	0.0056	0.00075961	0.0113	0.00075961
SE(n)	0.1666		0.3333
SE(a)	0.0062		0.0062
AIC	-56.5514		-56.5514
CAIC	-55.8455		-55.8455
BIC	-54.5599		-54.5599
HQIC	-56.1626		-56.1626
LL	30.2757		30.2757
K-S Value	0.0977		0.0977
H0	Fail to reject		Fail to reject
P-value	0.9458		0.9458
AD	0.3728		0.3728
CVM	0.0545		0.0545
Determinant	0.00039		0.0016
Significant(n)	0		0
Significant(a)	0		0

Analysis shows that although the metrics (AIC, CAIC, BIC & HQIC) for MBUR and Unit Lindley are slightly higher than those of the GOMBUR-1 but the determinant for the latter is less than the variance of the MBUR and the variance of the Unit Lindley indicating more efficiency. Both MBUR and the generalized form fit the data better than the Beta distribution. The determinants of both Beta and Kumaraswamy are higher than the determinants of the generalized forms of MBUR. The determinant for GOMBUR-1 is lesser than the determinant for the GOMBUR-2 which suggests better efficiency of GOMBUR-1 over the GOMBUR-2. Fitting both GOMBUR-1 & GOMBUR-2 yields the same estimate of alpha, variance and standard error. Both parameters of the generalized forms are highly significant.

Table 4 shows the result of analysis of the educational attainment dataset.

The analysis shows that the Unit Lindley fits the data well, followed by the MBUR then the Kumaraswamy and lastly comes the Beta distribution. Fitting GOMBUR-1 and 2 comes before the Beta distribution for fitting the data. The striking news is that the determinants for both generalized forms are less than the variance of any distribution fitting the data which makes the generalized distributions more efficient for fitting the data to these distributions. Also, as in the previous data analysis, the determinant for GOMBUR-1 is less than the determinant for the GOMBUR-2. Alpha estimate after fitting MBUR distribution is nearly equal to its value after estimating the GOMBUR-1 & 2 . Figure 13 show the theoretical CDFs of the competitors while Figure 14 shows the fitted PDFs. Figure 15, Figure 16 and Figure 17 shows the PP plot and QQ plot for the tested distributions. Figure 18 shows the CDFs and PDFs of GOMBUR-1 and GOMBUR-2.

Table 4. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=0.9014$		$n=2.8027$
	$\alpha =0.5531$		$\alpha =0.5531$
Variance	0.1775	0.0046	0.7102	0.0092
	0.0046	0.0013	0.0092	0.0013
SE(n)	0.0702		0.1405
SE(a)	0.006		0.006
AIC	-46.9242		-46.9242
CAIC	-46.5606		-46.5606
BIC	-43.7572		-43.7572
HQIC	-45.8189		-45.8189
LL	25.4621		25.4621
K-S Value	0.1421		0.1421
H0	Fail to reject		Fail to reject
P-value	0.2228		0.2228
AD	1.1827		1.1827
CVM	0.1957		0.1957
Determinant	0.00020661		0.00082644
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 4. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=0.9014$		$n=2.8027$
	$\alpha =0.5531$		$\alpha =0.5531$
Variance	0.1775	0.0046	0.7102	0.0092
	0.0046	0.0013	0.0092	0.0013
SE(n)	0.0702		0.1405
SE(a)	0.006		0.006
AIC	-46.9242		-46.9242
CAIC	-46.5606		-46.5606
BIC	-43.7572		-43.7572
HQIC	-45.8189		-45.8189
LL	25.4621		25.4621
K-S Value	0.1421		0.1421
H0	Fail to reject		Fail to reject
P-value	0.2228		0.2228
AD	1.1827		1.1827
CVM	0.1957		0.1957
Determinant	0.00020661		0.00082644
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

The CDF of the GOMBUR-1 perfectly aligns with the CDFs of the Beta, the Kumaraswamy and the MBUR distributions; this can be deduced from the approximate values of AIC, CAIC, BIC and HQIC. This is also true for the fitted PDFs as shown in Figure 14. Fitted CDFs and fitted PDF s of both GOMBURT-1 and GOMBURT-2 are identical as illustrated in Figure 18 and this has a reflection on the equal indices ( AIC, CAIC, BIC, HQIC, AD, CVM, and KS statistics) obtained from the analysis. Also, the alignment seen in PP plot depicted in Figure 17 for both versions is identical.

Table 5 shows the result of flood data analysis

The analysis shows that Beta distribution fits the data better than any other distribution. The Topp Leone did not fit the distribution. Generalization of MBUR using the GOMBUR-1 improves the fitting up to the level of the Beta distribution and slightly exceeding it. Marked increases in the negativity levels of AIC, CAIC, BIC & HQIC are obtained. The level of Log-likelihood shows marked improvement. Marked reduction in the levels of AD and CVM statistics are obvious. The variance of alpha shows marked reduction after fitting the GOMBUR-1. The determinant of GOMBUR-1&2 is far less than the determinant of the Beta distribution and Kumaraswamy distribution demonstrating more efficiency. GOMBUR-1 has lesser determinant than the GOMBUR-2. The estimates of alpha value, their variances and standard errors are identical for both versions. While the estimates of the n parameter, its variance and standard error obtained after fitting GOMBUR-2 is higher than the levels obtained after fitting GOMBUR-1 distribution. Figure 19 shows near perfect alignment of the theoretical CDF of GOMBUR-1 distribution with the Beta distribution. This is also evident in Figure 20 which portrays near perfect alignment of fitted PDFs for both distributions.

Table 5. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=8.1044$		$n=17.2087$
	$\alpha =1.1168$		$\alpha =1.1168$
Variance	8.0302	0.0177	32.1208	0.0354
	0.0177	0.0018	0.0354	0.0018
SE(n)	0.6336		1.2673
SE(a)	0.0095		0.0095
AIC	-24.4562		-24.4562
CAIC	-23.7503		-23.7503
BIC	-22.4647		-22.4647
HQIC	-24.0674		-24.0674
LL	14.2281		14.2281
K-S Value	0.204		0.204
H0	Fail to reject		Fail to reject
P-value	0.3297		0.3297
AD	0.7153		0.7153
CVM	0.1205		0.1205
Determinant	0.0143		0.0573
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 5. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=8.1044$		$n=17.2087$
	$\alpha =1.1168$		$\alpha =1.1168$
Variance	8.0302	0.0177	32.1208	0.0354
	0.0177	0.0018	0.0354	0.0018
SE(n)	0.6336		1.2673
SE(a)	0.0095		0.0095
AIC	-24.4562		-24.4562
CAIC	-23.7503		-23.7503
BIC	-22.4647		-22.4647
HQIC	-24.0674		-24.0674
LL	14.2281		14.2281
K-S Value	0.204		0.204
H0	Fail to reject		Fail to reject
P-value	0.3297		0.3297
AD	0.7153		0.7153
CVM	0.1205		0.1205
Determinant	0.0143		0.0573
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Both Figure 19 & Figure 20 reveal how MBUR exhibits near perfect alignment with the beta distribution after using the generalized version. Figure 21 elucidates the PP plot and QQ plot for the competitor distributions. The plots for Beta and Kumaraswamy suggest better fit of the data. Figure 22 demonstrates the amendment in alignment of the diagonal after fitting the GOMBUR-1 in dissimilarity to the alignment with the fitted MBUR. Figure 23 exemplifies the PP plot of the GOMBUR-1 & 2. They are identical. Moreover, Figure 24 illuminates the fitted CDFs and the fitted PDFs for both versions of the generalization expounding identity of the curves.

Table 6 shows the results of time between failures data analysis

The analysis shows that the all the proposed distributions fit the data except the Unit Lindley distributions. The GOMBURT-1 fits the data with slight negativity reduction in some indices like AIC, BIC & HQIC and enhanced reduction in AD,CVM and KS tests which points a good remark for adding a mentionable improvement in fitting. The Log-likelihood value obtained with generalization is higher than that obtained after fitting MBUR and exceeding slightly the levels obtained after fitting the Beta and Kumaraswamy distributions. Also, the variance of the alpha after fitting the GOMBUR-1 is less than the variance obtained after fitting the MBUR and all other distributions. The Kumaraswamy distribution has the least determinant among kumaraswamy, GOMBUR-1 & 2. But other indices like AIC, CAIC, BIC & HQIC make the generalization a better alternative. It is a matter of tradeoff between efficiency and biasness. The determinant of the variance- covariance matrix obtained after fitting the Beta distribution is slightly larger than that obtained after GOMBUR-2 fitting. Moreover, the latter has better indices than the indices of the Beta distribution. All this explains the enhanced parameter estimation when generalizing the MBUR distribution by adding a new shape parameter.

Table 6. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.8592$		$n=4.7184$
	$\alpha =1.8362$		$\alpha =1.8362$
Variance	0.6516	0.0251	2.6064	0.0502
	0.0251	0.143	0.0502	0.0143
SE(n)	0.1683		0.3366
SE(a)	0.0249		0.0249
AIC	-37.3005		-37.3005
CAIC	-36.7005		-36.7005
BIC	-35.0295		-35.0295
HQIC	-36.7293		-36.7293
LL	20.6502		20.6502
K-S Value	0.1355		0.1355
H0	Fail to reject		Fail to reject
P-value	0.7424		0.7424
AD	0.5399		0.5399
CVM	0.0897		0.0897
Determinant	0.0087		0.0348
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 6. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.8592$		$n=4.7184$
	$\alpha =1.8362$		$\alpha =1.8362$
Variance	0.6516	0.0251	2.6064	0.0502
	0.0251	0.143	0.0502	0.0143
SE(n)	0.1683		0.3366
SE(a)	0.0249		0.0249
AIC	-37.3005		-37.3005
CAIC	-36.7005		-36.7005
BIC	-35.0295		-35.0295
HQIC	-36.7293		-36.7293
LL	20.6502		20.6502
K-S Value	0.1355		0.1355
H0	Fail to reject		Fail to reject
P-value	0.7424		0.7424
AD	0.5399		0.5399
CVM	0.0897		0.0897
Determinant	0.0087		0.0348
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Figure 25 shows a proximate alignment of the GOMBUR-1 with the Beta, Kumaraswamy, MBUR, and Topp Loene distributions especially at the ends of the distributions and most of the center of these distributions. No such alignment is well-defined with the Unit Lindley distribution as the latter does not essentially fit the data. Figure 26 displays the same outcome. Figure 27 spectacles the QQ plot and PP plot for different competitors. Figure 28 exposes how the GOMBUR-1 enhances the lower tail alignment with the diagonal over the MBUR. Figure 29 and Figure 30 clear up that GOMBUR-1 &2 have the same PP plots, the same fitted CDFs and the same fitted PDFs with right skewness.

Table 7 shows the results of analysis of COVID-19 death rate analysis in Canada.

The analysis shows that both the Beta distribution and the Kumaraswamy distribution fit the data well while MBUR, Topp Leone and Unit Lindley distributions did not fit the data. The GOMBUR-1&2 fit the data with levels of AIC, CAIC, BIC & HQIC far exceeding the MBUR, although slightly less than the Kumaraswamy and Beta distribution. The advantages of fitting the generalized form of MBUR is that generalization reduces the variance as illustrated by the markedly reduced values of the determinants of the variance-covariance matrices obtained after fitting the GOMBUR-1&2 distributions. The estimated variance for the alpha is markedly less than the variance of the estimated alpha obtained from Beta and Kumaraswamy fitting . The estimated alpha levels, their variances and standard errors are the same after fitting both versions of GOMBUR distributions. Figure 31 and Figure 32 show perfect alignment of Beta distribution and GOMBUR-1. Kumaraswamy fitting has more negative values of AIC, BIC & HQIC indicating outperformance over both Beta and GOMBUR-1 data fitting, but in the same time it has a high value of the determinant of the variance-covariance matrix which yields a less efficient fitting of data.

Table 7. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=41.02961$		$n=83.0593$
	$\alpha =1.4623$		$\alpha =1.4623$
Variance	62.6379	0.0083	250.5451	0.0166
	0.0083	0.0002376	0.0166	0.00023762
SE(n)	1.0576		2.1152
SE(a)	0.0021		0.0021
AIC	-167.5661		-167.5661
CAIC	-167.3396		-167.3396
BIC	-163.5154		-163.5154
HQIC	-165.9956		-165.9956
LL	85.783		85.783
K-S Value	0.0781		0.0781
H0	Fail to reject		Fail to reject
P-value	0.6461		0.6461
AD	0.4653		0.4653
CVM	0.0728		0.0728
Determinant	0.0148		0.0593
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 7. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=41.02961$		$n=83.0593$
	$\alpha =1.4623$		$\alpha =1.4623$
Variance	62.6379	0.0083	250.5451	0.0166
	0.0083	0.0002376	0.0166	0.00023762
SE(n)	1.0576		2.1152
SE(a)	0.0021		0.0021
AIC	-167.5661		-167.5661
CAIC	-167.3396		-167.3396
BIC	-163.5154		-163.5154
HQIC	-165.9956		-165.9956
LL	85.783		85.783
K-S Value	0.0781		0.0781
H0	Fail to reject		Fail to reject
P-value	0.6461		0.6461
AD	0.4653		0.4653
CVM	0.0728		0.0728
Determinant	0.0148		0.0593
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Figure 33 shows graphs of PP plot and QQ plot for the competitor distributions. Figure 34 illuminates how generalization of MBUR markedly enhances the diagonal alignment of GOMBUR-1 along both ends and along the center of the distribution. Figure 35 shows the PP plots of both versions of the distribution being identical. Figure 36 depicts the fitted CDFs and the fitted PDFs of both versions. The shapes of the PDFs are symmetric and identical for both versions. And this is anticipated from the large values of the estimated n, the larger the estimated n is, the more symmetrical the distribution is.

Table 8 shows the results of COVID-19 death rates in Spain

The analysis demonstrates that the Beta and Kumaraswamy distributions fit the data well as shown from the AIC, CAIC, BIS &HQIC. The MBUR, Topp Loene and Unit Lindley distributions did not fit the data. But the generalized form of MBUR in the form of GOMBUR-1 & 2 FIT THE DATA BETTER with more negative values for all the mentioned indices, lowered values for AD, CVM & KS statistics, and a larger Log-Likelihood value. This in addition with lower values of the determinants of the variance-covariance matrices in comparison with the values obtained of fitting Beta and Kumaraswamy distributions. Generalization renders the GOMBUR-1 &2 fitting the data more efficient than the Beta or Kumaraswamy fitting. Figure 37 illuminates the alignments of the theoretical CDFs of Beta and GOMBUR-1, both run in parallel with Kumaraswamy CDF. Figure 38 shows the same orientation. Figure 39 displays the PP plot and the QQ plot of the competitors. Figure 40 shows the improvement of the diagonal alignment of GOMBUR as an effect of generalizing the BMUR distribution which exhibit improper alignment. Figure 41 presents the identical PP plots for both versions of the GOMBUR. Figure 42 illustrates the similarity of the CDFs and PDFs of both versions. The PDF curve shows mild right skewness.

Table 8. : to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=10.7533$		$n=22.5065$
	$\alpha =1.3801$		$\alpha =1.3801$
Variance	4.0831	0.0066	16.3323	0.0133
	0.0066	0.00065056	0.0133	0.00065056
SE(n)	0.2487		0.4975
SE(a)	0.0031		0.0031
AIC	-111.7925		-111.7925
CAIC	-111.6020		-111.6020
BIC	-107.4132		-107.4132
HQIC	-110.062		-110.062
LL	57.8962		57.8962
K-S Value	0.1105		0.1105
H0	Fail to reject		Fail to reject
P-value	0.3689		0.3689
AD	1.0013		1.0013
CVM	0.168		0.168
Determinant	0.0026		0.0104
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 8. : to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=10.7533$		$n=22.5065$
	$\alpha =1.3801$		$\alpha =1.3801$
Variance	4.0831	0.0066	16.3323	0.0133
	0.0066	0.00065056	0.0133	0.00065056
SE(n)	0.2487		0.4975
SE(a)	0.0031		0.0031
AIC	-111.7925		-111.7925
CAIC	-111.6020		-111.6020
BIC	-107.4132		-107.4132
HQIC	-110.062		-110.062
LL	57.8962		57.8962
K-S Value	0.1105		0.1105
H0	Fail to reject		Fail to reject
P-value	0.3689		0.3689
AD	1.0013		1.0013
CVM	0.168		0.168
Determinant	0.0026		0.0104
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 9 shows the result of analysis of the COVID-19 death rates in United Kingdom

The analysis illustrates that the Kumaraswamy and Beta fit the data well while the MBUR, Topp Leaone and Unit Lindley do not fit the data. The GOMBUR-1& 2 fit the data better than the MBUR. It is the third distribution after the Beta and Kumaraswamy as if has the third negative values of AIC, CAIC, BIC & HQIC after Kumaraswamy and Beta distribution. But it has the lesser values of the determinant of the variance covariance matrix obtained after estimating the distribution in comparison with fitting the Beta and the Kumaraswamy distribution. Figure 43 shows the theoretical CDFs of the fitted distribution. The CDF of the GOMBUR-1 is perfectly aligned with the CDF of the Beta distribution. Figure 44 shows similar orientation but for the fitted PDFs. Figure 45 shows the PP plot and QQ plot for the competitor distributions. Figure 45 demonstrates the enhanced effects of generalization and how it improves the diagonal alignment of the GOMBUR-1 &2 than the alignment of MBUR. Figure 46 illuminates the similar PP plot for the two versions and lastly Figure 48 shows the similar fitted CDF and PDF for the two versions.

Table 9. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=8.1471$		$n=17.2952$
	$\alpha =1.3616$		$\alpha =1.3616$
Variance	2.7020	0.0072	10.8082	0.0144
	0.0072	0.00090006	0.0144	0.00090006
SE(n)	0.2122		0.4244
SE(a)	0.0039		0.0039
AIC	-86.6033		-86.6033
CAIC	-86.3928		-86.3928
BIC	-82.4146		-82.4146
HQIC	-84.9649		-84.9649
LL	45.3017		45.3017
K-S Value	0.0971		0.0971
H0	Fail to reject		Fail to reject
P-value	0.5896		0.5896
AD	0.7562		0.7562
CVM	0.1319		0.1319
Determinant	0.0024		0.0095
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 9. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=8.1471$		$n=17.2952$
	$\alpha =1.3616$		$\alpha =1.3616$
Variance	2.7020	0.0072	10.8082	0.0144
	0.0072	0.00090006	0.0144	0.00090006
SE(n)	0.2122		0.4244
SE(a)	0.0039		0.0039
AIC	-86.6033		-86.6033
CAIC	-86.3928		-86.3928
BIC	-82.4146		-82.4146
HQIC	-84.9649		-84.9649
LL	45.3017		45.3017
K-S Value	0.0971		0.0971
H0	Fail to reject		Fail to reject
P-value	0.5896		0.5896
AD	0.7562		0.7562
CVM	0.1319		0.1319
Determinant	0.0024		0.0095
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 10 shows the result of analysis of the petroleum rock dataset

The analysis revealed that the Beta and Kumaraswamy distributions fitted the data well. MBUR , Topp Leaone and Unit Lindley did not fit the data. The GOMBUR-1& 2 fitted the data with values of AIC, CAIC, BIC, HQIC far exceeding the MBUR fitting. Both generalized forms outperform the Beta and the Kumaraswamy fitting. Both versions of generalization have higher value of the Log-likelihood than any other distribution. Moreover, the lowest value of the determinant of the variance covariance matrix after fitting the GOMBER-1 distribution renders it the most efficient distribution among all distributions. It also has the lower values of the test statistics AD, CVM & KS. Figure 49 shows the perfect alignment of GOMBUR-1 and Beta CDFs curves along with the Kumaraswamy CDF. This pattern is also obvious for the fitted PDFs in Figure 50. The PP plots and the QQ plots are shown in Figure 51 while in Figure 52 the enhanced alignment of the diagonal is spectacle after fitting the GOMBUR-1 than after fitting the MBUR. In Figure 53, the similarity of PP plots between the two versions of generalization is recognizable. Figure 54 discloses the theoretical CDFs and fitted PDFs of GOMBUR-1& 2 which are similar to each other.

Table 10. to be continued:.

	GOMBUR-1		GOMBUR-2
theta	$n=6.7991$		$n=14.5982$
	$\alpha =1.469$		$\alpha =1.469$
Variance	2.4435	0.0097	9.7742	0.0194
	0.0097	0.0015	0.0194	0.0015
SE(n)	0.2256		0.4513
SE(a)	0.0057		0.0057
AIC	-68.1668		-68.1668
CAIC	-67.9002		-67.9002
BIC	-64.4244		-64.4244
HQIC	-66.7526		-66.7526
LL	36.0834		36.0834
K-S Value	0.1688		0.1688
H0	Fail to reject		Fail to reject
P-value	0.1152		0.1152
AD	2.7045		2.7045
CVM	0.4554		0.4554
Determinant	0.0037		0.0147
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 10. to be continued:.

	GOMBUR-1		GOMBUR-2
theta	$n=6.7991$		$n=14.5982$
	$\alpha =1.469$		$\alpha =1.469$
Variance	2.4435	0.0097	9.7742	0.0194
	0.0097	0.0015	0.0194	0.0015
SE(n)	0.2256		0.4513
SE(a)	0.0057		0.0057
AIC	-68.1668		-68.1668
CAIC	-67.9002		-67.9002
BIC	-64.4244		-64.4244
HQIC	-66.7526		-66.7526
LL	36.0834		36.0834
K-S Value	0.1688		0.1688
H0	Fail to reject		Fail to reject
P-value	0.1152		0.1152
AD	2.7045		2.7045
CVM	0.4554		0.4554
Determinant	0.0037		0.0147
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 11 shows the result of analysis of the snow fall dataset

The analysis discloses that the Beta, Kumaraswamy , MBUR and Unit Lindley distributions fit the data well but Topp Leone does not fit the data. The GOMBUR-1 & 2 fit the data better than any other distributions because both generalized forms have the highest negative values of AIC, CAIC, BIC & HQIC . They also have the highest Log-Likelihood value among them. In addition, they have the lowest values of the following tests: AD, CVM & KS. However, GOMBUR-1 has the lowest value of the determinant of variance covariance matrix. This solidifies GOMBUR-1 the most efficient distribution to fit the data well. Figure 55 illuminates the impeccable alignment of the theoretical CDF curves of the Beta and Kumaraswamy. The MBUR properly aligns with the two curves mainly at the lower end while the Unit Lindley unimpeachably aligns with the same curves mainly at the upper end. However, the GOMBUR-1 distribution faultlessly aligns along the ends and along the center of the two curves. Figure 56 exhibit the same manner. Figure 57 illustrates the PP plots and QQ plots for the competitor distributions.

Table 11. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=4.6735$		$n=10.3469$
	$\alpha =1.9954$		$\alpha =1.9954$
Variance	2.043	0.021	8.172	0.042
	0.021	0.0063	0.042	0.0063
SE(n)	0.261		0.5219
SE(a)	0.0145		0.0145
AIC	-77.3883		-77.3883
CAIC	-76.9439		-76.9439
BIC	-74.5889		-74.5889
HQIC	-76.4918		-76.4918
LL	40.6942		40.6942
K-S Value	0.1011		0.1011
H0	Fail to reject		Fail to reject
P-value	0.8889		0.8889
AD	0.3453		0.3453
CVM	0.0443		0.0443
Determinant	0.0125		0.0498
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 11. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=4.6735$		$n=10.3469$
	$\alpha =1.9954$		$\alpha =1.9954$
Variance	2.043	0.021	8.172	0.042
	0.021	0.0063	0.042	0.0063
SE(n)	0.261		0.5219
SE(a)	0.0145		0.0145
AIC	-77.3883		-77.3883
CAIC	-76.9439		-76.9439
BIC	-74.5889		-74.5889
HQIC	-76.4918		-76.4918
LL	40.6942		40.6942
K-S Value	0.1011		0.1011
H0	Fail to reject		Fail to reject
P-value	0.8889		0.8889
AD	0.3453		0.3453
CVM	0.0443		0.0443
Determinant	0.0125		0.0498
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Figure 58 high-lights the enhancing effect of generalization in sense of proper diagonal alignment at both ends and center after fitting the GOMBUR-1 in comparison with MBUR, although the latter distribution fits the data. Figure 59 depicts the resemblance of the PP plots of both versions of GOMBUR. Figure 60 emphasizes the indistinguishable curves of theoretical CDFs and fitted PDFs of both versions. The PDF curve is highly right skewed.

Table 12 shows the result of analysis of the milk production dataset

The analysis revealed that all the distributions fit the data. But the Unit Lindley is the best among them followed by Kumaraswamy, Beta , Topp Leone and lastly BMUR distribution. Unit Lindley has the highest negative values of AIC, CAIC, BIC & HQIC. The case is not true in sense of having the lowest values of AD, CVM & KS. But, Kumaraswamy distribution has the lowest values of AD, CVM & KS and the highest Log-likelihood value that is slightly larger than that of the Unit Lindley. The two parameters Kumaraswamy have, may account for the reduction in AIC, CAIC, BIC, & HQIC in comparison with the Unit Lindley which has one parameter. The GOMBUR 1 & 2 increases the log-likelihood value. The generalization increases the negativity of the AIC, CAIC, BIC, & HQIC, lower the AD, CVM & KS and increases log-likelihood value. Also, the generalization lowers the variance of the alpha parameter among all distribution. The determinant of the variance covariance matrix obtained after fitting GOMBUR-1 is the lowest among all distributions having two parameters. It is a matter of convenience to choose any distributions according to the need of the research. While the Unit Lindley is the best to fit the data, MBUR is the worst to fit the data. Generalization enhances its capability for fitting the data to approach the level of the Beta distribution fitting the data.

Table 12. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.7903$		$n=4.5807$
	$\alpha =1.0644$		$\alpha =1.0664$
Variance	0.1327	0.0031	0.531	0.0061
	0.0031	0.0011	0.0061	0.0011
SE(n)	0.0352		0.0704
SE(a)	0.0031		0.0031
AIC	-43.262		-43.262
CAIC	-43.1466		-43.1466
BIC	-37.9163		-37.9163
HQIC	-41.0949		-41.0949
LL	23..631		23..631
K-S Value	0.0803		0.0803
H0	Fail to reject		Fail to reject
P-value	0.3366		0.3366
AD	1.367		1.367
CVM	0.2191		0.2191
Determinant	0.00013161		0.00052642
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 12. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=1.7903$		$n=4.5807$
	$\alpha =1.0644$		$\alpha =1.0664$
Variance	0.1327	0.0031	0.531	0.0061
	0.0031	0.0011	0.0061	0.0011
SE(n)	0.0352		0.0704
SE(a)	0.0031		0.0031
AIC	-43.262		-43.262
CAIC	-43.1466		-43.1466
BIC	-37.9163		-37.9163
HQIC	-41.0949		-41.0949
LL	23..631		23..631
K-S Value	0.0803		0.0803
H0	Fail to reject		Fail to reject
P-value	0.3366		0.3366
AD	1.367		1.367
CVM	0.2191		0.2191
Determinant	0.00013161		0.00052642
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Figure 61 illustrates the theoretical CDFs of different competitors. Figure 62 discloses the fitted PDFs for the same distributions. Figure 63 depict PP plots and QQ plots for the competitors. Figure 64 exposes the PP plots of MBUR and the GOMBUR-1. Figure 65 exemplifies the PP plots for the both versions. Figure 66 elucidates the fitted PDFs and CDFs of both versions.

Table 13 shows analysis results of COVID-19 recovery rate in Spain dataset.

The analysis exposes that the Kumaraswamy followed by Beta distributions are the distributions that fir the data well. This is because they have the best indices. GOMBUR-1&2 have indices better than that of the Beta rendering the generalized versions the second distribution to fit the after Kumaraswamy. In advantage, GOMBUR-1 has the least variance of alpha among all the distributions and has the least value of the determinant of variance covariance matrix. This makes it the most efficient distribution to fit the data. Also the covariance between the two parameters in the generalized form is far less than that obvious in Kumaraswamy and Beta distributions. Figure 67 and Figure 68 reveal the theoretical CDFs and fitted PDFs for the competitors. Figure 69 uncovers the PP plots and QQ plots for the competitors. Figure 70, Figure 71 and Figure 72 display the PP plot of the MBUR, GOMBUR-1 & 2, the fitted PDFs and CDFs for the two generalized versions.

Table 13. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=4.0506$		$n=9.1012$
	$\alpha =0.6705$		$\alpha =0.6705$
Variance	0.7317	0.0032	2.9267	0.0064
	0.0032	0.00036616	0.0064	0.00036616
SE(n)	0.1053		0.2106
SE(a)	0.0024		0.0024
AIC	-111.3714		-111.3714
CAIC	-111.1809		-111.1809
BIC	-106.9921		-106.9921
HQIC	-109.641		-109.641
LL	57.6857		57.6857
K-S Value	0.0977		0.0977
H0	Fail to reject		Fail to reject
P-value	0.3436		0.3436
AD	1.0324		1.0324
CVM	0.1743		0.1743
Determinant	0.00025776		0.001
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 13. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=4.0506$		$n=9.1012$
	$\alpha =0.6705$		$\alpha =0.6705$
Variance	0.7317	0.0032	2.9267	0.0064
	0.0032	0.00036616	0.0064	0.00036616
SE(n)	0.1053		0.2106
SE(a)	0.0024		0.0024
AIC	-111.3714		-111.3714
CAIC	-111.1809		-111.1809
BIC	-106.9921		-106.9921
HQIC	-109.641		-109.641
LL	57.6857		57.6857
K-S Value	0.0977		0.0977
H0	Fail to reject		Fail to reject
P-value	0.3436		0.3436
AD	1.0324		1.0324
CVM	0.1743		0.1743
Determinant	0.00025776		0.001
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 14 shows analysis results of voter dataset.

The analysis reveals the Beta followed by Topp Leone, Kumaraswamy, MBUR and lastly Unit Lindley distributions fit the data well. This conclusion is guided by the indices as illustrated in Table 14. GOMBUR-1, in comparison to MBUR, has more negative values of AIC, CAIC, BIC & HQIC. It also has lower values of AD, CVM & KS. In addition, it has higher Log-likelihood value than that of the MBUR. All these enhancement of the indices renders the GOMBUR-1 to be the second distribution to fit the data after the Beta distribution. Taking into account that GOMBUR-1 has the least variance of the estimated alpha among all the distributions and has far less value of determinant of its estimated variance covariance matrix than that of the Beta distribution; this leverages it to be the most efficient distribution to fit the data. The covariance between the parameters of GOMBUR-1 is far less than that evident between the parameters of the Beta distribution. Figure 73 and Figure 74 describe the theoretical CDFs and the fitted PDFs of the competitors respectively. Figure 75 clarifies the PP plots and QQ plots of the competitors. Figure 76 and Figure 77 depict the comparison of the PP plots between MBUR, GOMBUR-1 & 2. Figure 78 explicates the fitted PDFs and CDFs of the two versions of generalization.

Table 14. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=2.9469$		$n=6.8938$
	$\alpha =0.713$		$\alpha =0.713$
Variance	0.7663	0.0059	3.0651	0.0118
	0.0059	0.0009294	0.0118	0.0009294
SE(n)	0.142		0.284
SE(a)	0.0049		0.0049
AIC	-47.8278		-47.8278
CAIC	-47.4849		-47.4849
BIC	-44.5526		-44.5526
HQIC	-46.6625		-46.6625
LL	25.9139		25.9139
K-S Value	0.0949		0.0949
H0	Fail to reject		Fail to reject
P-value	0.8519		0.8519
AD	0.2797		0.2797
CVM	0.0428		0.0428
Determinant	0.00067749		0.0027
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 14. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=2.9469$		$n=6.8938$
	$\alpha =0.713$		$\alpha =0.713$
Variance	0.7663	0.0059	3.0651	0.0118
	0.0059	0.0009294	0.0118	0.0009294
SE(n)	0.142		0.284
SE(a)	0.0049		0.0049
AIC	-47.8278		-47.8278
CAIC	-47.4849		-47.4849
BIC	-44.5526		-44.5526
HQIC	-46.6625		-46.6625
LL	25.9139		25.9139
K-S Value	0.0949		0.0949
H0	Fail to reject		Fail to reject
P-value	0.8519		0.8519
AD	0.2797		0.2797
CVM	0.0428		0.0428
Determinant	0.00067749		0.0027
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 15 shows analysis results of unit capacity factors dataset.

The analysis revealed that the Kumaraswamy followed by beta , Topp Leone and lastly BMUR distributions fit the data well as indicated by the indices. The GOMBUR-1 distribution also fit the data with better indices. It has slightly larger negative values of AIC, CAIC, BIC, & HQIC and minimally larger value of Log-Likelihood than those of the Kumaraswamy distribution. It has a marginally lower value of KS than that of the Kumaraswamy distribution. The variance of the estimated alpha after fitting the Kumaraswamy distribution is less than that after the GOMBUR-1. The determinant of the variance covariance matrix obtained after fitting the Kumaraswamy distribution is less than that after fitting the GOMBUR-1 which justifies it to be more efficient distribution to fit the data rather than GOMBUR distribution. Figure 79 and Figure 80 show the fitted CDFs and PDF s of the competitors. Figure 81, Figure 82 and Figure 83 illustrates the PP plot of the different competitors. Figure 84 displays the fitted CDFs and PDFs of the GOMBUR-1 & 2 distributions.

Table 15. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=0.1959$		$n=1.3917$
	$\alpha =1.5298$		$\alpha =1.5298$
Variance	0.1087	0.0214	0.435	0.0428
	0.0214	0.0258	0.0428	0.0258
SE(n)	0.0688		0.1375
SE(a)	0.0335		0.0335
AIC	-15.365		-15.365
CAIC	-14.765		-14.765
BIC	-13.094		-13.094
HQIC	-14.7939		-14.7939
LL	9.6825		9.6825
K-S Value	0.1782		0.1782
H0	Fail to reject		Fail to reject
P-value	0.4103		0.4103
AD	0.7026		0.7026
CVM	0.1161		0.1161
Determinant	0.0024		0.0094
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

Table 15. to be continued.

	GOMBUR-1		GOMBUR-2
theta	$n=0.1959$		$n=1.3917$
	$\alpha =1.5298$		$\alpha =1.5298$
Variance	0.1087	0.0214	0.435	0.0428
	0.0214	0.0258	0.0428	0.0258
SE(n)	0.0688		0.1375
SE(a)	0.0335		0.0335
AIC	-15.365		-15.365
CAIC	-14.765		-14.765
BIC	-13.094		-13.094
HQIC	-14.7939		-14.7939
LL	9.6825		9.6825
K-S Value	0.1782		0.1782
H0	Fail to reject		Fail to reject
P-value	0.4103		0.4103
AD	0.7026		0.7026
CVM	0.1161		0.1161
Determinant	0.0024		0.0094
Significant(n)	P<0.001		P<0.001
Significant(a)	P<0.001		P<0.001

4. Conclusion

As illustrated from the analysis of the datasets adding new parameters to the MBUR increases its flexibility to accommodate different shapes of data with different characteristics like skewness , kurtosis and different tail behaviors. This newly introduced parameter improves the estimation process by increasing validity indices like AIC, CAIC, BIS & HQIC. It also enhances the goodness of fit by reducing test statistics like AD, CVM & KS tests. It also increases the value of the Log-Likelihood. The two versions of the generalization yield different values for the n parameter but equal values for the alpha parameter. The variances of the estimated alpha obtained from the two versions are identical. The covariance between the two parameters is minimal. It is far less than the covariance obtained from fitting distributions like the Beta and the Kumaraswamy. The determinant of the estimated variance covariance matrix obtained from fitting GOMBUR-1 is minimal and almost with a least values in comparison with that obtained after fitting Beta and Kumaraswamy distribution to the data. There is an exception for this that is obvious in the unit capacity factors. The determinant of the matrix obtained after fitting the Kunmaraswamy distribution is less than that obtained from the GOMBUR-1. Otherwise, there is a pattern of minimal value of the determinant of the matrix obtained by GOMBUR fitting to data. It is a matter of tradeoff to weigh the benefits of increased validity indices against increased efficiency of estimators which has a minimal variance. Generalization renders the MBUR perfectly aligns with the Beta distribution and more or less parallels the Kumaraswamy in most datasets.

In this paper Nelder Mead optimizer was used for MLE estimation of the parameters. Bayesian inference procedures may be used in future works.