Bayesian analysis of series system with dependent causes of failure

Figures & data

Table 1. Possible reference prior of (α, θ₁, θ₂, θ₃).

Ordered	Reference prior	Reference prior for
grouping	for (α, θ1, θ2, θ3)	(α, λ0, λ1, λ2)
{(α, θ1), (θ2, θ3)}	ω1(α, θ1, θ2, θ3)	π1(α, λ0, λ1, λ2)
{(θ2, θ3), (α, θ1)}	ω1(α, θ1, θ2, θ3)	π1(α, λ0, λ1, λ2)
{(θ2, θ3), α, θ1}	ω1(α, θ1, θ2, θ3)	π1(α, λ0, λ1, λ2)
{α, (θ1, θ2, θ3)}	ω1(α, θ1, θ2, θ3)	π1(α, λ0, λ1, λ2)
{α, θ1, θ2, θ3}	ω2(α, θ1, θ2, θ3)	π2(α, λ0, λ1, λ2)
{θ3, θ2, θ1, α}	ω3(α, θ1, θ2, θ3)	π3(α, λ0, λ1, λ2)
{θ1, (α, θ2, θ3)}	ω4(α, θ1, θ2, θ3)	π4(α, λ0, λ1, λ2)
{(θ2, θ3), θ1, α}	ω4(α, θ1, θ2, θ3)	π4(α, λ0, λ1, λ2)

Table 2. The RB(%), CPs and RMSEs of the parameters when (α, λ₀, λ₁, λ₂) = (0.5, 0.7, 1, 1.5).

n	Method		α	θ1	θ2	θ3
	MLE	RMSE	0.1337	1.8909	0.1196	0.1241
		RB(%)	9.9284	10.4440	0.1600	2.5209
		CP	0.9480	0.9710	0.8730	0.9410
	ω1	RMSE	0.1310	2.0187	0.1088	0.1147
		RB(%)	9.0203	11.2454	0.7515	4.9180
		CP	0.9400	0.9300	0.9560	0.9600
15	ω2	RMSE	0.1310	2.0187	0.1128	0.1165
		RB(%)	9.0203	11.2454	3.8310	6.2795
		CP	0.9400	0.9300	0.9490	0.9550
	ω3	RMSE	0.1248	1.7886	0.1084	0.1160
		RB(%)	7.5734	11.2454	0.7515	4.9180
		CP	0.9490	0.9530	0.9560	0.9460
	ω4	RMSE	0.1248	1.7886	0.1088	0.1147
		RB(%)	7.5734	11.2454	0.7515	4.9180
		CP	0.9490	0.9530	0.9560	0.9600
	MLE	RMSE	0.0857	0.9554	0.0871	0.0922
		RB(%)	5.1869	9.8592	0.5333	0.8213
		CP	0.9470	0.9640	0.9040	0.9340
	ω1	RMSE	0.0845	0.9650	0.0830	0.0878
		RB(%)	4.6945	9.6570	0.1905	0.5934
		CP	0.9400	0.9400	0.9430	0.9580
30	ω2	RMSE	0.0845	0.9650	0.0844	0.0882
		RB(%)	4.6945	9.6570	1.4378	1.3578
		CP	0.9400	0.9400	0.9370	0.9440
	ω3	RMSE	0.0828	0.9139	0.0830	0.0894
		RB(%)	4.0475	9.6570	0.1905	0.5934
		CP	0.9430	0.9450	0.9290	0.9470
	ω4	RMSE	0.0828	0.9139	0.0830	0.0878
		RB(%)	4.0475	9.6570	0.1905	0.5934
		CP	0.9430	0.9450	0.9430	0.9580
	MLE	RMSE	0.0614	0.6505	0.0663	0.0686
		RB(%)	3.1057	5.5929	0.0320	0.3648
		CP	0.9410	0.9660	0.9240	0.9610
	ω1	RMSE	0.0610	0.6559	0.0644	0.0666
		RB(%)	2.8219	5.4257	0.2252	0.4872
		CP	0.9340	0.9460	0.9570	0.9610
50	ω2	RMSE	0.0610	0.6559	0.0652	0.0669
		RB(%)	2.8219	5.4257	1.1799	0.9393
		CP	0.9340	0.9460	0.9530	0.9570
	ω3	RMSE	0.0601	0.6340	0.0643	0.0673
		RB(%)	2.4622	5.4257	0.2252	0.4872
		CP	0.9480	0.9470	0.9460	0.9560
	ω4	RMSE	0.0601	0.6340	0.0644	0.0666
		RB(%)	2.4622	5.4257	0.2252	0.4872
		CP	0.9480	0.9470	0.9570	0.9610

Table 3. The RB(%), CPs and SRMSEs of the parameters when (α, λ₀, λ₁, λ₂) = (1, 0.7, 1, 1.5).

n	Method		α	θ1	θ2	θ3
	MLE	RMSE	0.2711	1.9711	0.1186	0.1246
		RB(%)	9.7428	12.5952	0.7147	1.5538
		CP	0.9510	0.9630	0.8770	0.9140
	ω1	RMSE	0.2654	2.0968	0.1078	0.1147
		RB(%)	8.7483	14.8890	0.0436	4.0388
		CP	0.9500	0.9290	0.9540	0.9650
	ω2	RMSE	0.2654	2.0968	0.1121	0.1169
		RB(%)	8.7483	14.8890	3.2041	5.4434
15		CP	0.9500	0.9290	0.9480	0.9580
	ω3	RMSE	0.2517	1.9172	0.1082	0.1171
		RB(%)	7.3402	14.8890	0.0436	4.0388
		CP	0.9520	0.9380	0.9520	0.9550
	ω4	RMSE	0.2517	1.9172	0.1078	0.1147
		RB(%)	7.3402	14.8890	0.0436	4.0388
		CP	0.9520	0.9380	0.9540	0.9650
	MLE	RMSE	0.1587	0.8689	0.0848	0.0923
		RB(%)	3.5935	8.2261	0.8747	0.6436
		CP	0.9480	0.9650	0.9140	0.9130
	ω1	RMSE	0.1574	0.8757	0.0807	0.0879
		RB(%)	3.1080	7.9506	0.5156	0.7628
		CP	0.9440	0.9490	0.9500	0.9590
	ω2	RMSE	0.1574	0.8757	0.0822	0.0884
		RB(%)	3.1080	7.9506	1.0826	1.5091
30		CP	0.9440	0.9490	0.9490	0.9400
	ω3	RMSE	0.1539	0.8313	0.0807	0.0895
		RB(%)	2.4891	7.9506	0.5156	0.7628
		CP	0.9470	0.9500	0.9430	0.9440
	ω4	RMSE	0.1539	0.8313	0.0807	0.0879
		RB(%)	2.4891	7.9506	0.5156	0.7628
		CP	0.9470	0.9500	0.9500	0.9590
	MLE	RMSE	0.1116	0.6033	0.0636	0.0694
		RB(%)	2.2050	4.8884	0.3072	0.1813
		CP	0.9550	0.9540	0.9290	0.9580
	ω1	RMSE	0.1113	0.6034	0.0617	0.0675
		RB(%)	1.8847	4.6502	0.4924	0.6654
		CP	0.9520	0.9410	0.9560	0.9580
	ω2	RMSE	0.1113	0.6034	0.0625	0.0679
50		RB(%)	1.8847	4.6502	0.5682	1.1268
		CP	0.9520	0.9410	0.9550	0.9520
	ω3	RMSE	0.1098	0.5866	0.0618	0.0682
		RB(%)	1.5879	4.6502	0.4924	0.6654
		CP	0.9540	0.9380	0.9530	0.9490
	ω4	RMSE	0.1098	0.5866	0.0617	0.0675
		RB(%)	1.5879	4.6502	0.4924	0.6654
		CP	0.9540	0.9380	0.9560	0.9580

Table 4. The RB(%), CPs and RMSEs of the parameters when (α, λ₀, λ₁, λ₂) = (2, 0.7, 1, 1.5).

n	Method		α	θ1	θ2	θ3
	MLE	RMSE	0.5570	2.8778	0.1170	0.1265
		RB(%)	10.7944	10.9046	0.4160	1.2409
		CP	0.9440	0.9760	0.8820	0.9070
	ω1	RMSE	0.5308	2.7675	0.1064	0.1162
		RB(%)	9.6667	10.7426	0.2279	3.7543
		CP	0.9310	0.9410	0.9560	0.9600
	ω2	RMSE	0.5308	2.7675	0.1106	0.1177
		RB(%)	9.6667	10.7426	3.4277	5.1166
15		CP	0.9310	0.9410	0.9520	0.9500
	ω3	RMSE	0.5079	2.3783	0.1063	0.1187
		RB(%)	8.2546	10.7426	0.2279	3.7543
		CP	0.9430	0.9450	0.9570	0.9420
	ω4	RMSE	0.5079	2.3783	0.1064	0.1162
		RB(%)	8.2546	10.7426	0.2279	3.7543
		CP	0.9430	0.9450	0.9560	0.9600
	MLE	RMSE	0.3251	0.9553	0.0837	0.0878
		RB(%)	5.0828	9.9498	0.6400	0.4587
		CP	0.9600	0.9630	0.9210	0.9350
	ω1	RMSE	0.3215	0.9729	0.0797	0.0840
		RB(%)	4.5786	9.7440	0.2921	1.8125
		CP	0.9550	0.9440	0.9640	0.9600
	ω2	RMSE	0.3215	0.9729	0.0812	0.0846
		RB(%)	4.5786	9.7440	1.3427	2.5132
30		CP	0.9550	0.9440	0.9540	0.9570
	ω3	RMSE	0.3126	0.9069	0.0796	0.0849
		RB(%)	3.9656	9.7440	0.2921	1.8125
		CP	0.9570	0.9500	0.9510	0.9590
	ω4	RMSE	0.3126	0.9069	0.0797	0.0840
		RB(%)	3.9656	9.7440	0.2921	1.8125
		CP	0.9570	0.9500	0.9640	0.9600
	MLE	RMSE	0.2500	0.6531	0.0663	0.0739
		RB(%)	3.3462	5.7905	0.0960	0.2747
		CP	0.9460	0.9580	0.9280	0.9040
	ω1	RMSE	0.2476	0.6539	0.0644	0.0719
		RB(%)	3.0103	5.5583	0.2874	0.9139
		CP	0.9520	0.9460	0.9540	0.9470
50	ω2	RMSE	0.2476	0.6539	0.0652	0.0721
		RB(%)	3.0103	5.5583	1.2762	1.3791
		CP	0.9520	0.9460	0.9450	0.9380
	ω3	RMSE	0.2444	0.6322	0.0643	0.0725
		RB(%)	2.6582	5.5583	0.2874	0.9139
		CP	0.9480	0.9460	0.9440	0.9390
	ω4	RMSE	0.2444	0.6322	0.0644	0.0719
		RB(%)	2.6582	5.5583	0.2874	0.9139
		CP	0.9480	0.9460	0.9540	0.9470

Table 5. Minimum time to blindness in days and its causes for 71 patients with diabetic retinopathy.

i	T	δ*	i	T	δ*	i	T	δ*	i	T	δ*	i	T	δ*
1	266	1	16	125	2	31	717	2	46	663	0	61	503	1
2	91	2	17	777	2	32	642	1	47	567	2	62	423	2
3	154	2	18	306	1	33	141	2	48	966	0	63	285	2
4	285	0	19	415	1	34	407	1	49	203	0	64	315	2
5	583	1	20	307	2	35	356	1	50	84	1	65	727	2
6	547	2	21	637	2	36	1653	0	51	392	1	66	210	2
7	79	1	22	577	2	37	427	2	52	1140	2	67	409	2
8	622	0	23	178	1	38	699	1	53	901	1	68	584	1
9	707	2	24	517	2	39	36	2	54	1247	0	69	355	1
10	469	2	25	272	0	40	667	1	55	448	2	70	1302	1
11	93	1	26	1137	0	41	588	2	56	904	2	71	227	2
12	1313	2	27	1484	1	42	471	0	57	276	1
13	805	1	28	315	1	43	126	1	58	520	1
14	344	1	29	287	2	44	350	2	59	485	2
15	790	2	30	1252	1	45	350	1	60	248	2

Figure 1. The Weibull probability plot.

Table 6. The estimates of the parameters for the DRS data.

Method	α	95%CI	θ1	95%CI	θ2	95%CI	θ3	95%CP
MLE	1.558	(1.276,1.841)	0.469	(0.321,0.617)	0.394	(0.281,0.508)	0.465	(0.351,0.579)
ω1	1.557	(1.286,1.846)	0.471	(0.337,0.632)	0.393	(0.285,0.507)	0.462	(0.349,0.577)
ω2	1.557	(1.286,1.846)	0.471	(0.337,0.632)	0.396	(0.290,0.512)	0.460	(0.346,0.574)
ω3	1.549	(1.285,1.831)	0.477	(0.341,0.633)	0.391	(0.279,0.506)	0.464	(0.351,0.579)
ω4	1.549	(1.285,1.831)	0.477	(0.341,0.633)	0.393	(0.285,0.507)	0.462	(0.349,0.577)