Mc-Donald modified Burr-III distribution: properties and applications

Figures & data

Figure 1. PDF plot of the McMB–III distribution.

Figure 2 Hazard plot of the McMB–III distribution.

Figure 3. (a) Effect of α for fixed $\beta ,\gamma ,\lambda ,\eta$, and $\xi$values. (b) Effect of $\beta$ for fixed $\alpha ,\gamma ,\lambda ,\eta$, and $\xi$ values. (c) Effect of $\gamma$ for fixed $\alpha ,\beta ,\lambda ,\eta$, and $\xi$ values. (d) Effect of $\lambda$ for fixed $\alpha ,\beta ,\gamma ,\eta$, and $\xi$ values. (e) Effect of $\eta$ for fixed $\alpha ,\beta ,\gamma ,\lambda ,$ and $\xi$ values. (f) Effect of $\xi$ for fixed $\alpha ,\beta ,\gamma ,\lambda$, and $\eta$ values.

Table 1. Summary of some special cases of the McMB–III distribution.

	Parameters
Distribution	$\alpha$	$\beta$	$\gamma$	$\lambda$	$\eta$	$\xi$	Distribution Name
MB–III	$\alpha$	$\beta$	$\gamma$	1	1	1	Modified Burr–III distribution
B–III	$\alpha$	$\beta$	1	1	1	1	Burr–III distribution
McB–III	$\alpha$	$\beta$	1	$\lambda$	$\eta$	$\xi$	McDonald Burr–III distribution
BMB–III	$\alpha$	$\beta$	$\gamma$	$\lambda$	$\eta$	1	Beta modified Burr–III distribution
BBIII	$\alpha$	$\beta$	1	$\lambda$	$\eta$	1	Beta Burr–III distribution
KMBIII	$\alpha$	$\beta$	$\gamma$	1	$\eta$	$\xi$	Kumarasamy modified Burr–III distribution
KBIII	$\alpha$	$\beta$	1	1	$\eta$	$\xi$	Kumarasamy Burr–III distribution
EMBIII	$\alpha$	$\beta$	$\gamma$	1	1	$\xi$	Exponential modified Burr–III distribution
EBIII	$\alpha$	$\beta$	1	1	1	$\xi$	Exponential Burr–III distribution
MLL	1	$\beta$	1	$\lambda$	$\eta$	$\xi$	Modified Log–Logistic distribution
KLL	1	$\beta$	1	1	$\eta$	$\xi$	Kumarasamy Log–Logistic distribution
BLL	1	$\beta$	1	$\lambda$	$\eta$	1	Beta Log–Logistic distribution
LL	1	$\beta$	1	1	1	1	Log–Logistic distribution

Figure 4. (a) $(\beta =4.43,\gamma =1.5,\lambda =1.15,\eta =1,\xi =0.53)$. (b) $(\alpha =10,\gamma =1.5,\lambda =1,\eta =1,\xi =0.53)$. (c) $(\alpha =9.24,\beta =4.43,\lambda =1.148,\eta =1,\xi =0.53)$. (d) $(\alpha =10,\beta =4.43,\gamma =1.15,\eta =1,\xi =0.53)$. (e) $(\alpha =10,\beta =4.43,\gamma =1.5,\lambda =1.15,\xi =0.53)$. (f) $(\alpha =10,\beta =4.43,\gamma =1.5,\lambda =1.15,\eta =0.53)$.

Table 2. Maximum Likelihood Estimator and Information Criterion for breaking strengths of single carbon fibres.

	MLE						Information Criterion
Model	$\alpha$	$\beta$	$\gamma$	$\lambda$	$\eta$	$\xi$	W	AIC	CAIC	HQIC
McMBIII	3.106	1.558	0.00	3.046	13.889	3.164	112.7	124.7	126.1	129.8
MBIII	2043.21	7.198	1025.1	1	1	1	124.9	130.9	131.3	133.45
BIII	233.77	5.44	1	1	1	1	127.8	131.8	131.99	133.50
BMBIII	5.92	1.141	0.0001	6.05	15.10	1	112.62	122.62	123.65	119.75
KMBIII	4.988	1.775	0.000	1	17.17	4.57	113.107	123.107	124.14	120.23

Table 3. Maximum Likelihood Estimator and Information Criterion Average Annual Percent Change in Private Health Insurance Premiums.

	MLE							Information Criterion
Model	$\alpha$	$\beta$	$\gamma$	$\lambda$	$\eta$	$\xi$	W	AIC	BIC	CAIC	HQIC
McMBIII	22.73	1.46	28.002	0.244	23.40	11.68	224.894	236.894	246.72	239.60	240.38
BIII	29.05	1.69	1	1	1	1	242.292	246.292	249.57	257.0	247.46
KMBIII	26.14	1.13	53.19	1	46.91	5.37	225.141	231.141	236.05	239.85	232.89

Figure 5. Histogram & fitted distribution.

Figure 6. Histogram & fitted distribution.

Table

$\alpha =1,\beta =2,\gamma =1$
	$\xi =1$			$\xi =2$			$\xi =3$
Λ	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$
1	0.45	0.38	0.33	0.85	0.76	0.71	1.12	1.02	0.96
2	0.77	0.65	0.56	1.24	1.10	1.0	1.58	1.41	1.30
3	1.0	0.84	0.74	1.54	1.34	1.21	1.93	1.70	1.55
4	1.18	1.0	0.88	1.78	1.54	1.39	2.23	1.94	1.76
5	1.34	1.13	1.0	1.99	1.72	1.54	2.48	2.15	1.95
6	1.48	1.25	1.11	2.18	1.88	1.68	2.71	2.35	2.11

Table

$\alpha =2,\beta =1,\gamma =1$
	$\xi =1$			$\xi =2$			$\xi =3$
Λ	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$
1	0.41	0.35	0.32	1.28	1.15	1.05	2.16	1.95	1.81
2	1.12	0.91	0.78	2.71	2.28	2.01	4.30	3.66	3.25
3	1.80	1.44	1.21	4.08	3.34	2.88	6.36	5.25	4.56
4	2.47	1.95	1.63	5.43	4.37	3.72	8.39	6.80	5.83
5	3.14	2.46	2.04	6.78	5.39	4.55	10.41	8.34	7.07
6	3.81	2.96	2.45	8.12	6.14	5.37	12.43	9.86	8.30

Table

$\alpha =1,\beta =1,\gamma =2$
	$\xi =1$			$\xi =2$			$\xi =3$
Λ	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$	$\eta =2$	$\eta =3$	$\eta =4$
1	–	–	–	0.0	0.0	0.4	0.39	0.33	0.71
2	–	–	–	0.67	0.5	0.4	1.44	1.15	1.82
3	0.25	0.15	0.09	1.33	1.0	0.8	2.46	1.93	2.87
4	0.56	0.36	0.25	2.0	1.5	1.2	3.47	2.69	3.90
5	0.88	0.59	0.42	2.67	2.0	1.6	4.47	3.45	4.92
6	1.21	0.82	0.60	3.33	2.5	2.0	5.48	4.21	5.93

Table

Estimates	parameters	n = 50	n = 100	n = 150	n = 200
	0.5	1.1	0.4	0.5	0.6
	1.3	0.9	1.7	1.1	1.2
	1.7	0.5	0.6	1.7	1.75
	1.4	0.5	0.8	1.2	1.3
	0.65	0.6	0.4	0.7	0.6
	0.6	0.6	0.3	0.5	0.6
Log likelihood		−84	−172	−272	−280
P-values	Anderson-Darling	0.5	0.2	0.7	0.8
	Kolmogorov-Smirnov	0.6	0.4	0.6	0.7
	Pearson ${\chi }^2$	0.6	0.6	0.7	0.8