Applications of Burr III-Weibull quantile function in reliability analysis

Figures & data

Figure 1. Plots of the density function for various parameter values.

Figure 2. Plots of the hazard quantile function for various parameter values.

Table 1. The sample L-moment values for Dataset-1 and Dataset-2.

	$l_1$	$l_2$	$l_3$	$l_4$
Dataset-1	4.3253	0.5688	−0.0500	0.0919
Dataset-2	92.0745	51.3599	22.0285	11.062

Table 2. The basic descriptive statistics of the two datasets.

	Median	IQR	Skewness (S)	Kurtosis (T)
Dataset-1	4.3566	1.3232	−0.0358	1.246
Dataset-2	55.1305	107.198	0.3477	1.4241

Table 3. Goodness-of-fit statistics corresponding to the two datasets.

	Dataset-1			Dataset-2
	Parameters	${\chi }^2$(dof)	p-value	Parameters	${\chi }^2$(dof)	p-value
LSE	$c=7.084$	19.2516(13)	0.115	$c=2.2448$	16.6332(16)	0.409
	$k=0.5882$			$k=16.8983$
	$\lambda =6.2665$			$\lambda =0.8285$
	$\sigma =3.8404$			$\sigma =82.7944$
MLM	$c=7.4363$	9.3154(13)	0.748	$c=1.9101$	11.8295(16)	0.7556
	$k=0.4457$			$k=16.9201$
	$\lambda =5.6163$			$\lambda =0.7971$
	$\sigma =3.7564$			$\sigma =78.9684$

Figure 3. $Q\text{-}Q$ plot corresponing to LSE and MLM estimates for the first dataset.

Figure 4. $Q\text{-}Q$ plot corresponing to LSE and MLM estimates for the second dataset.

Table 4. The chi-square with p-values for the datasets.

	Dataset-1		Dataset-2
Models	${\chi }^2$	p-value	${\chi }^2$	p-value
BIIIW	9.31543	0.7487	11.8295	0.7556
W	12.4163	0.4938	14.4481	0.5653
BIII	18.8007	0.1294	21.5262	0.1591

Figure 5. (a) Hazard quantile plot of the first dataset. (b) Hazard quantile plot of the second dataset.

Figure 6. (a) Mean residual plot of the first dataset. (b) Mean residual plot of the second dataset.

Figure 7. (a) The fitted density of the first dataset. (b) The fitted density of the second dataset.

Table A1. Fracture toughness data of Alumina $\left(\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\right)$.

5.5	5	4.9	6.4	5.1	5.2	5.2	5	4.7	4
4.5	4.2	4.1	4.56	5.01	4.7	3.13	3.12	2.68	2.77
2.7	2.36	4.38	5.73	4.35	6.81	1.91	2.66	2.61	1.68
2.04	2.08	2.13	3.8	3.73	3.71	3.28	3.9	4	3.8
4.1	3.9	4.05	4	3.95	4	4.5	4.5	4.2	4.55
4.65	4.1	4.25	4.3	4.5	4.7	5.15	4.3	4.5	4.9
5	5.35	5.15	5.25	5.8	5.85	5.9	5.75	6.25	6.05
5.9	3.6	4.1	4.5	5.3	4.85	5.3	5.45	5.1	5.3
5.2	5.3	5.25	4.75	4.5	4.2	4	4.15	4.25	4.3
3.75	3.95	3.51	4.13	5.4	5	2.1	4.6	3.2	2.5
4.1	3.5	3.2	3.3	4.6	4.3	4.3	4.5	5.5	4.6
4.9	4.3	3	3.4	3.7	4.4	4.9	4.9	5

Table A2. Number of failures for the air conditioning system of jet airplanes.

194	413	90	74	55	23	97	50	359	50
130	487	57	102	15	14	10	57	320	261
51	44	9	254	493	33	18	209	41	58
60	48	56	87	11	102	12	5	14	14
29	37	186	29	104	7	4	72	270	283
7	61	100	61	502	220	120	141	22	603
35	98	54	100	11	181	65	49	12	239
14	18	39	3	12	5	32	9	438	43
134	184	20	386	182	71	80	188	230	152
5	36	79	59	33	246	1	79	3	27
201	84	27	156	21	16	88	130	14	118
44	15	42	106	46	230	26	59	153	104
20	206	5	66	34	29	26	35	5	82
31	118	326	12	54	36	34	18	25	120
31	22	18	216	139	67	310	3	46	210
57	76	14	111	97	62	39	30	7	44
11	63	23	22	23	14	18	13	34	16
18	130	90	163	208	1	24	70	16	101
52	208