Statistical analysis of dependent competing risks model in constant stress accelerated life testing with progressive censoring based on copula function

Figures & data

Figure 1. Scatter plots of BMP with different Kendall's tau.

Table 1. The pre-fixed sampling scheme.

Case	n1	r1		n2	r2		n3	r3
I	20	6	(3,2,2,3,2,2)	40	12	(3,2,2,…, 3,2,2)	80	24	(3,2,2,…, 3,2,2)
II	20	12	(1,1,0,…1,1,0)	40	24	(1,1,0,…1,1,0)	80	48	(1,1,0,…1,1,0)

Table 2. Average MLEs, MSEs, BPCIs, BTCIs and CPs under different stress levels in two cases when (τ₁₁, τ₁₂, τ₂₁, τ₂₂, τ₃₁, τ₃₂) = (6.8395, 7.4135, 4.5542, 4.5148, 3.2266, 2.9657), (m₁, m₂) = (2, 1.8), θ = 1.

Testing plan	Case I			Case II
Parameters	MLEs (MSEs)	BPCIs CPs	BTCIs CPs	MLEs (MSEs)	BPCIs CPs	BTCIs CPs
τ11	7.2510 (1.1125)	(4.5621, 9.5124) 0.9270	(3.9894, 9.2511) 0.9350	6.9211 (1.1054)	(4.6785, 9.5211) 0.9520	(3.9032, 8.4577) 0.9460
τ12	7.3589 (1.1547)	(4.4325, 9.8014) 0.9370	(3.7854, 8.9602) 0.9420	7.4874 (1.0201)	(4.6987,9.0003) 0.9580	(3.1245, 8.3211) 0.9610
τ21	4.6720 (0.8965)	(2.9685, 6.1714) 0.9250	(1.4214, 6.2387) 0.9460	4.5966 (0.8108)	(2.8857, 6.0032) 0.9600	(1.7754, 5.2018) 0.9440
τ22	4.4547 (0.8487)	(2.6598, 6.1469) 0.9350	(2.0521, 5.8715) 0.9340	4.4987 (0.7856)	(2.6098, 6.0122) 0.9550	(1.5987, 5.2705) 0.9450
τ31	3.4369 (0.7544)	(1.4102, 4.5368) 0.9310	(0.8547, 3.9874) 0.9260	3.3022 (0.7091)	(1.6987, 4.0789) 0.9520	(0.8074, 4.0221) 0.9460
τ32	3.3897 (0.8217)	(1.1478, 4.3698) 0.9340	(0.8854, 3.6597) 0.9400	3.1986 (0.6855)	(1.2587, 4.0538) 0.9320	(0.0024, 3.9608) 0.9480
m1	2.2147 (0.3698)	(1.3864, 3.1153) 0.9380	(0.5960, 2.4712) 0.9490	1.9849 (0.2098)	(1.5584, 2.4001) 0.9510	(0.9806, 2.2644) 0.9450
m2	1.9025 (0.3001)	(1.1054, 2.4987) 0.9460	(0.7785, 2.0956) 0.9440	1.8547 (0.1935)	(1.2547, 2.3365) 0.9340	(0.8857, 2.2579) 0.9590
θ	1.1125 (0.4035)	(0.5986,1.4335) 0.9120	(0.6784,1.5698) 0.9200	1.0986 (0.3051)	(0.4725,1.1989) 0.9350	(0.0035,1.2966) 0.9410

Table 3. Estimators of acceleration coefficients in two cases when (a₁, a₂, b₁, b₂) = ( − 3, −4, 1640, 2000), θ = 1.

Testing plan	Parameters	a1	a2	b1	b2
Case I	MLE	−3.5687	−4.6520	1877	2204
	MSE	1.5406	1.8744	502	621
Case II	MLE	−3.4322	−4.1279	1774	2078
	MSE	1.1254	1.4511	442	549

Table 4. Average MLEs, MSEs, BPCIs, BTCIs and CPs under different stress levels in two cases when (τ₁₁, τ₁₂, τ₂₁, τ₂₂, τ₃₁, τ₃₂) = (6.8395, 7.4135, 4.5542, 4.5148, 3.2266, 2.9657), (m₁, m₂) = (2, 1.8), θ = 2.

Testing plan	Case I			Case II
Parameters	MLEs (MSEs)	BPCIs CPs	BTCIs CPs	MLEs (MSEs)	BPCIs CPs	BTCIs CPs
τ11	7.1145 (1.4571)	(4.2510, 9.7225) 0.9350	(1.7985, 8.3328) 0.9270	7.0504 (1.2589)	(4.3396, 9.0957) 0.9410	(1.8500, 8.7336) 0.9500
τ12	7.6637 (1.6987)	(4.0327, 10.8998) 0.9400	(1.7985, 8.5698) 0.9430	7.4478 (1.1250)	(4.6587, 9.9965) 0.9360	(2.3321,8.9633) 0.9420
τ21	4.7666 (0.9718)	(2.9605, 6.7854) 0.9290	(1.4533, 5.9632) 0.9410	4.6547 (0.9411)	(3.0958, 6.1579) 0.9570	(1.9658, 5.3366) 0.9600
τ22	4.8041 (0.8741)	(2.9687, 6.7441) 0.9490	(1.5784, 5.7789) 0.9300	4.5987 (0.7985)	(3.0052,5.6985) 0.9480	(1.2258, 5.4369) 0.9620
τ31	3.5625 (0.7554)	(1.4874, 4.9052) 0.9340	(0.5698, 3.8471) 0.9260	3.2987 (0.6985)	(1.6548, 4.7752) 0.9440	(0.8890, 3.9961) 0.9350
τ32	3.2598 (0.7114)	(1.1784, 4.3598) 0.9360	(0.4747, 3.9652) 0.9380	2.8741 (0.6621)	(1.1056, 3.8714) 0.9540	(0.5269, 3.2458) 0.9470
m1	2.2544 (0.4521)	(1.0537, 3.2210) 0.9420	(0.3524, 2.6754) 0.9370	1.8896 (0.3002)	(1.1025, 2.6524) 0.9550	(1.0024, 2.1108) 0.9480
m2	1.9587 (0.3002)	(1.1574, 2.3305) 0.9430	(0.8451, 2.2732) 0.9320	1.7099 (0.2457)	(0.9762, 2.2487) 0.9470	(0.6785, 2.1017) 0.9510
θ	2.1132 (0.4322)	(1.2587,2.9645) 0.9250	(0.9352,2.6147) 0.9330	1.9365 (0.2874)	(1.4520,2.7855) 0.9420	(0.8695,2.3369) 0.9380

Table 5. Estimators of acceleration coefficients in two cases when (a₁, a₂, b₁, b₂) = ( − 3, −4, 1640, 2000), θ = 2.

Testing plan	Parameters	a1	a2	b1	b2
Case I	MLE	−3.5471	−4.6300	1799	2214
	MSE	1.4578	1.8957	549	633
Case II	MLE	−3.3369	−4.4722	1687	2143
	MSE	1.2547	1.5897	451	578

Figure 2. Comparison of three reliability functions when θ = 1.

Figure 3. Comparison of three reliability functions when θ = 2.

Table 6. MLEs, BPCIs and BTCIs of unknown parameters.

Parameters	MLEs	BPCIs	BTCIs
τ11	1437	(896, 2033)	(678, 1901)
τ12	2549	(1344, 3908)	(1058, 3569)
τ21	1278	(960, 1672)	(804, 1578)
τ22	1931	(1158, 2930)	(908, 2631)
τ31	783	(502, 1116)	(433, 1027)
τ32	872	(638, 1305)	(529, 1192)
m1	89.04	(61.77, 115.69)	(58.74, 104.13)
m2	116.90	(78.52, 160.24)	(69.88, 148.71)
θ	1.4320	(1.2580, 1.6323)	(1.1982, 1.5411)