Analysis of uncertainty measure using unified hybrid censored data with applications

Figures & data

Figure 1. Unified hybrid censoring scheme.

Table 1. MLE and Bayesian estimate of Shannon entropy based on UHCS under balanced loss functions ($\alpha =1.2,\varphi =0.55,H_0\left(x\right)=1.053$) and $\omega =0.5$ and MSE of all estimates.

				$(T_1=1.8$ $T_2=2.4)$
n	r	k	MLE	BSLE	BLINEX		BGE
					$(-0.5)$	$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$
60	50	20	1.04716	1.16646	1.17619	1.15713	1.15745	1.14036
			0.05833	0.05533	0.05565	0.05587	0.05604	0.05811
		30	1.0387	1.15659	1.16622	1.14736	1.14781	1.13115
			0.0564	0.0526	0.05316	0.05434	0.05301	0.05454
		40	0.99326	1.14227	1.15264	1.13234	1.13231	1.11366
			0.0533	0.04838	0.04828	0.04877	0.04983	0.05339
100	80	40	1.1271	1.12154	1.13014	1.19317	1.19319	1.1171
			0.04984	0.02984	0.02941	0.03046	0.03088	0.0335
		50	1.01844	1.11918	1.12642	1.1212	1.11223	1.09881
			0.04624	0.02891	0.0288	0.02918	0.02959	0.03138
		65	1.0129	1.11683	1.12434	1.10953	1.10964	1.09578
			0.03914	0.02719	0.02708	0.02748	0.02787	0.02975
		75	0.998	1.11026	1.11754	1.10318	1.10344	1.09029
			0.03184	0.01929	0.0193	0.01943	0.01968	0.02086
					$(T_1=2.1$	$T_2=5)$
60	50	20	1.0375	1.17586	1.18616	1.16598	1.16592	1.14735
			0.0692	0.05912	0.05945	0.0596	0.06048	0.06427
		30	1.03401	1.17171	1.18155	1.16228	1.16274	1.14571
			0.0652	0.05232	0.05207	0.05286	0.05271	0.05425
		40	1.01297	1.15334	1.16331	1.14378	1.14406	1.12644
			0.0622	0.05084	0.05095	0.05101	0.05165	0.05394
100	80	40	1.1138	1.14589	1.15323	1.13873	1.13867	1.1947
			0.04531	0.03812	0.0377	0.0387	0.04914	0.05165
		50	1.01844	1.13537	1.14306	1.1279	1.12819	1.11431
			0.04131	0.03774	0.03186	0.0328	0.03812	0.03934
		65	0.97334	1.09361	1.10193	1.08551	1.08552	1.06988
			0.03521	0.02812	0.0277	0.0297	0.02914	0.02965
		75	0.92248	1.10219	1.10948	1.09509	1.09523	1.08179
			0.03121	0.02774	0.02776	0.02988	0.02812	0.02934
				$(T_1=3$  $T_2=5)$
n	r	k	MLE	BSLE	BLINEX		BGE
					$(-0.5)$	$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$
60	50	20	1.03694	1.17464	1.18438	1.1653	1.16556	1.14846
			0.0666	0.05726	0.0578	0.05802	0.05803	0.06025
		30	1.01893	1.16114	1.17114	1.15154	1.15195	1.13453
			0.0632	0.05062	0.05103	0.05252	0.05115	0.05294
		40	1.0154	1.16142	1.17146	1.15179	1.15199	1.13416
			0.0611	0.04764	0.04789	0.04973	0.04856	0.05134
100	80	40	1.10269	1.13407	1.1422	1.13605	1.13607	1.19062
			0.0374	0.031	0.03961	0.0457	0.03198	0.03393
		50	1.01969	1.12265	1.13	1.11549	1.11556	1.10187
			0.0324	0.0308	0.03063	0.03112	0.03154	0.03353
		65	1.0119	1.11764	1.12493	1.11054	1.11071	1.09733
			0.0224	0.02701	0.02692	0.02723	0.02757	0.0291
		75	1.01005	1.11771	1.12527	1.11034	1.11049	1.09656
			0.0214	0.02771	0.02657	0.02699	0.02737	0.0286
					$(T_1=3$	$T_2=8)$
60	50	20	1.16436	1.16715	1.17599	1.1587	1.15945	1.14482
			0.05917	0.05417	0.05525	0.05332	0.05387	0.05475
		30	1.08436	1.13265	1.14346	1.12229	1.12262	1.10357
			0.0531	0.0398	0.03992	0.04005	0.04059	0.04308
		40	1.03472	1.11487	1.11532	1.10485	1.11512	1.01665
			0.044	0.0317	0.03188	0.03281	0.0388	0.04311
100	80	40	1.01475	1.12811	1.13552	1.1209	1.12103	1.10739
			0.0336	0.02833	0.02824	0.02857	0.02899	0.03073
		50	1.0117	1.11406	1.12201	1.10633	1.10654	1.09206
			0.0314	0.02714	0.02703	0.02742	0.02773	0.02936
		65	1.0028	1.00268	1.00986	1.00568	1.10109	1.1011
			0.0236	0.02417	0.02451	0.02497	0.0252	0.0256
		75	0.97503	1.08964	1.0974	1.0821	1.08208	1.06749
			0.0221	0.02339	0.02298	0.02395	0.0244	0.02694

Table 2. MLE and Bayesian estimate of Shannon entropy based on UHCS under balanced loss functions ($\alpha =0.9,\varphi =3.55,H_0\left(x\right)=3.48342$) and $\omega =0.5$ and MSE of all estimates.

				$(T_1=1.8$ $T_2=2.4)$
n	r	k	MLE	BSLE	BLINEX		BGE
					$(-0.5)$	$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$
60	50	20	3.42778	3.36145	3.36696	3.35573	3.35975	3.35628
			0.11347	0.10306	0.10218	0.10398	0.10349	0.10439
		30	3.44387	3.37546	3.38088	3.36981	3.37379	3.37039
			0.10045	0.09206	0.09147	0.0927	0.09239	0.09306
		40	3.45732	3.38604	3.39152	3.38033	3.38438	3.38099
			0.10005	0.09156	0.09135	0.0918	0.0917	0.092
100	80	40	3.43191	3.33119	3.33658	3.32558	3.32953	3.32613
			0.08041	0.08393	0.08297	0.08498	0.0843	0.08507
		50	3.45027	3.34688	3.35237	3.34116	3.34519	3.34172
			0.07594	0.07743	0.07647	0.07847	0.07783	0.07866
		65	3.44844	3.34463	3.35007	3.33895	3.34296	3.33955
			0.07493	0.07529	0.07453	0.07614	0.07559	0.07621
		75	3.43518	3.33554	3.34126	3.3296	3.33372	3.33003
			0.07377	0.07039	0.06859	0.07233	0.07117	0.07278
					$(T_1=2.1$	$T_2=5)$
60	50	20	3.4438	3.37482	3.38032	3.36909	3.37313	3.3697
			0.10963	0.10033	0.12984	0.10087	0.10062	0.10122
		30	3.45326	3.3831	3.38864	3.37733	3.38141	3.37796
			0.10847	0.09807	0.09967	0.09852	0.09834	0.09888
		40	3.43661	3.36885	3.37442	3.36305	3.36711	3.36357
			0.10483	0.09572	0.09471	0.09683	0.09623	0.09729
100	80	40	3.44302	3.34062	3.34606	3.33495	3.33894	3.33551
			0.09694	0.08322	0.0823	0.08422	0.08358	0.08433
		50	3.43337	3.33447	3.33978	3.32893	3.33283	3.32948
			0.08528	0.08145	0.07986	0.08318	0.08215	0.0836
		65	3.43657	3.33684	3.34244	3.33101	3.33506	3.33146
			0.08317	0.07869	0.07764	0.07986	0.07914	0.08009
		75	3.44776	3.34591	3.35145	3.34013	3.34419	3.34068
			0.08217	0.07709	0.07612	0.07916	0.0785	0.07933
				$(T_1=3$ $T_2=5)$
n	r	k	MLE	BSLE	BLINEX		BGE
					$(-0.5)$	$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$
60	50	20	3.45935	3.37438	3.38013	3.36842	3.37257	3.36889
			0.11287	0.10342	0.10237	0.10456	0.10394	0.105
		30	3.42156	3.35783	3.36345	3.35201	3.35609	3.35253
			0.10641	0.10261	0.10228	0.10298	0.10284	0.10332
		40	3.45246	3.38306	3.3886	3.37729	3.38137	3.37793
			0.10341	0.10218	0.10092	0.10152	0.10267	0.10228
100	80	40	3.44776	3.34565	3.3511	3.33997	3.34397	3.34053
			0.07687	0.07775	0.07679	0.07882	0.07816	0.079
		50	3.42886	3.32859	3.33406	3.3229	3.32688	3.3234
			0.07585	0.07638	0.07509	0.0767	0.0769	0.0788
		65	3.45399	3.35033	3.35585	3.34457	3.34862	3.34515
			0.07442	0.07386	0.073	0.07582	0.07423	0.07598
		75	3.44895	3.34613	3.35175	3.34027	3.34437	3.3408
			0.07134	0.06963	0.07145	0.07392	0.07215	0.07324
					$(T_1=3$	$T_2=8)$
60	50	20	3.42627	3.35906	3.36471	3.35318	3.35731	3.35374
			0.12587	0.11397	0.11311	0.1149	0.11443	0.11537
		30	3.41264	3.34484	3.34039	3.33906	3.34313	3.35396
			0.10809	0.10357	0.10233	0.10488	0.1042	0.10452
		40	3.42963	3.3653	3.37093	3.35946	3.36353	3.35994
			0.1057	0.09971	0.09902	0.10048	0.10006	0.10079
100	80	40	3.44236	3.33474	3.34018	3.32907	3.32307	3.31966
			0.0718	0.0745	0.07368	0.07541	0.07482	0.07547
		50	3.41426	3.31006	3.32548	3.3144	3.3184	3.315
			0.06946	0.07243	0.07165	0.0733	0.07273	0.07335
		65	3.43116	3.32954	3.33483	3.32403	3.32791	3.32458
			0.06094	0.06691	0.06599	0.06792	0.06725	0.06796
		75	3.43077	3.32984	3.33535	3.32411	3.32813	3.32466
			0.05274	0.06307	0.06047	0.06117	0.0605	0.06141

Figure 2. The MSE of Shannon entropy estimation for different values of sample size.

Figure 3. The MSE of Shannon entropy BEs based on BLINEX and BGE loss functions.

Figure 4. The MSE of Shannon entropy estimation under different values of k at n = 60.

Figure 5. The MSE of Shannon entropy estimates at ($T_1=2.1$ and $T_2=5$), ($T_1=3$ and $T_2=5$) and ($T_1=3$ and $T_2=8$) for $n=100,k=40$.

Figure 6. MCMC plots at $n=100,T_1=3,T_2=8,m=65,r=80$.

Figure 7. MCMC plots at $n=60,T_1=3,T_2=8,m=20,r=50$.

Table

0.96	4.15	0.19	0.78	8.01	31.75	7.35	6.50	8.27	33.91
32.52	3.16	4.85	2.78	4.67	1.31	12.06	36.71	72.89

Figure 8. Estimated pdf and cdf of the Lo distribution for first data.

Table 3. Shannon entropy estimators for first data.

Case	n	r	k	($T_1$, $T_2$)	MLE	BSLE	BLINEX		BGE
							$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$	$(-0.5)$
1		12	10		3.5991	3.03634	2.9408	3.12474	2.93568	3.00388
2		15	13		4.0366	3.56213	3.47079	3.64255	3.48013	3.53606
3	19	18	14	(20,35)	3.5928	3.03346	2.93892	3.12088	2.93377	3.00132
4		17	15		3.7010	3.16048	3.06747	3.24536	3.06623	3.13023
5		18	15		3.7107	3.16606	3.07176	3.25215	3.07069	3.13544
6		19	18		3.5991	3.01452	2.91166	3.10952	2.90488	2.97922

Figure 9. Estimated pdf and cdf of Lo distribution for second data.

Figure 10. The trace plot and histogram of posterior samples for case 1 of UHCS for the first real data.

Figure 11. The trace plot and histogram of posterior samples for case 1 of UHCS for the second real data.

Table 4. Shannon entropy estimators for second data.

Case	n	r	k	($T_1$, $T_2$)	MLE	BSLE	BLINEX		BGE
							$\left(0.5\right)$	$(-0.5)$	$\left(0.5\right)$	$(-0.5)$
1		25	27		6.83432	6.56384	6.5238	6.59975	6.54564	6.55789
2		25	30		6.81259	6.56198	6.51939	6.54533	6.44697	6.46245
3	52	25	35	(120,200)	6.79319	6.3732	6.3137	6.42777	6.34518	6.364
4		29	31		6.92354	6.55417	6.50037	6.60263	6.52961	6.54612
5		29	36		6.86165	6.36148	6.28443	6.43302	6.32491	6.34945
6		35	37		6.94537	6.44828	6.37005	6.52036	6.4117	6.43625