Chen-G class of distributions

Figures & data

Figure 1. Plots of density and hazard rate functions of CB distribution

Figure 2. Plots of the density and hazard rate function of CK distribution

Figure 3. Plots of density and hazard rate function of CW distribution

Table 1. Monte Carlo simulation results

n	Parameter	I			II
		AB	RMSE	CP	AB	RMSE	CP
50	$\lambda$	−0.5854	1.0708	0.9987	0.4737	0.9182	0.9913
	β	3.8663	56.4303	0.9977	2.6341	5.2212	0.9990
	α	−0.1005	0.1836	0.9977	0.6564	1.4488	0.9180
	γ	−0.0171	2.4805	0.9327	0.0876	0.3030	0.9920
150	$\lambda$	−0.2607	1.1688	0.9793	0.5179	0.9948	0.9873
	β	0.6373	1.2615	0.9945	1.6927	2.3435	0.9990
	α	−0.0534	0.1321	0.9867	0.6499	1.2680	0.9600
	γ	−0.1023	1.7291	0.9360	0.0652	0.1958	0.9927
300	$\lambda$	−0.1324	1.2618	0.9607	0.5254	1.0150	0.9793
	β	0.4988	0.9901	0.9973	1.5134	1.8010	0.9067
	α	−0.0396	0.1125	0.9853	0.5978	1.1114	0.9727
	γ	−0.2452	1.2307	0.9393	0.0484	0.1303	0.9913
600	$\lambda$	−0.0231	1.1910	0.9592	0.4924	1.0072	0.9580
	β	0.3936	0.5929	0.9900	1.4374	1.5874	0.7500
	α	−0.0240	0.0950	0.9633	0.5487	1.0468	0.9793
	γ	−0.2420	0.9657	0.9433	0.0405	0.1034	0.9827
1000	$\lambda$	0.0428	1.1763	0.9367	0.4089	0.8572	0.9393
	β	0.3599	0.5053	0.9640	1.3880	1.4766	0.6780
	α	−0.0173	0.0856	0.9407	0.4867	0.9565	0.9747
	γ	−0.2526	0.8181	0.9367	0.0402	0.0934	0.9513

Table 2. Fatigue time of 101 6061-T6 aluminum coupons

10	33	44	56	59	72	74	77	92	93	96	100
100	102	105	107	107	108	108	108	109	112	113	115
116	120	121	122	122	124	130	134	136	139	144	146
153	159	160	163	163	168	171	172	176	183	195	196
197	202	213	215	216	222	230	231	240	245	251	253
254	255	278	293	327	342	347	361	402	432	458	555

Table 3. Survival times of guinea pigs injected with different amounts of tubercle bacilli

70	90	96	97	99	103	104	104	105	107	108	108	108	109
109	112	112	113	114	114	114	116	119	120	120	120	121	121
123	124	124	124	124	124	128	128	129	129	130	130	130	131
131	131	131	131	132	132	132	133	134	134	134	134	136	136
137	138	138	138	139	139	141	141	142	142	142	142	142	142
142	142	144	144	145	146	148	148	149	151	151	152	155	156
155	156	157	157	157	157	158	159	162	163	163	164	166	166
168	170	174	201	212

Figure 4. TTT-transform plots for the datasets

Table 4. Maximum likelihood estimates and standard errors of parameters in brackets

Application	Model	$\hat{\lambda }$	$\hat{\beta }$	$\hat{\alpha }$	$\hat{\gamma }$	$\hat{\theta }$	$\hat{a}$	$\hat{b}$
Data1	CW	6.775	35.209	49.29	1.019
		(4.592)	(2.817)	(2.575)	(0.167)
	KEC		0.324	0.021		4.473	4.336	2.286
			(0.014)	(0.005)		(0.676)	(0.698)	(1.557)
	EC	0.289	0.245	1236.1
		(0.039)	(0.008)	$\left(4.15\times {10}^{-6}\right)$
	EW	0.020	55.14	1.493
		(0.001)	($7.27\times {10}^{-4}$)	(0.106)
Data 2	CW	19.366	15.742	30.945	0.31
		(36.009)	(1.242)	(12.491)	(0.098)
	KE-Chen		0.116	0.449		149.569	0.192	0.589
			(0.038)	(0.412)		($2.87\times {10}^{-4}$)	(0.234)	(0.006)
	E-Chen	0.865	0.138	163.36
		(0.077)	(0.008)	($1.79\times {10}^{-4}$)
	EW	4.626		242.255	0.272
		(2.378)		(0.094)	(0.022)

Table 5. Goodness-of-fit statistics and information criteria

Application	Model	KS	CM	AD	AIC	BIC	CAIC	HQIC
Data1	CW	0.064	0.046	0.299	901.984	912.365	902.41	906.184
	KEC	0.061	0.047	0.323	904.436	917.411	905.081	909.686
	EW	0.096	0.134	0.783	907.241	915.026	907.494	910.391
	EC	0.11	0.194	1.127	913.19	920.975	913.442	916.34
Data2	CW	0.092	0.09	0.564	859.704	868.81	860.301	863.329
	KEC	0.093	0.094	0.581	861.604	872.987	862.513	866.136
	EW	0.145	0.232	1.603	877.894	884.724	878.247	880.613
	EC	0.124	0.153	1.089	869.634	876.464	869.987	872.354

Figure 5. Empirical and fitted density and cdf plots of data1

Figure 6. Empirical and fitted density and cdf plots of data2

Figure 7. P-P plots of fitted distributions for data1

Figure 8. P-P plots of fitted distributions for data2

Figure 9. Profile log-likelihood plot of CW parameters for data1