Poisson–logarithmic half-logistic distribution with inference under a progressive-stress model based on adaptive type-II progressive hybrid censoring

Hashem, Atef F.; Kuş, Coşkun; Pekgör, Ahmet; Abdel-Hamid, Alaa H.

doi:10.1186/s42787-022-00148-9

Journal of the Egyptian Mathematical Society

Table 3 MLEs and PEs of \(\theta , p, \mu\) and \(\delta\) with their MSEs, RABs, AMSE, and ARAB based on 1000 simulations. Population parameter values are \(\theta =1.5\), \(p=0.9\), \(\mu =0.7\) and \(\delta =0.3\)

From: Poisson–logarithmic half-logistic distribution with inference under a progressive-stress model based on adaptive type-II progressive hybrid censoring

						MLE				PE
						\(\overline{\hat{\theta }}\)	MSE(\(\hat{\theta }\))	RAB(\(\hat{\theta }\))		\(\overline{\check{\theta }}\)	MSE(\(\check{\theta }\))	RAB(\(\check{\theta }\))
						\(\overline{\hat{p}}\)	MSE(\(\hat{p}\))	RAB(\(\hat{p}\))		\(\overline{\check{p}}\)	MSE(\(\check{p}\))	RAB(\(\check{p}\))
		\(n_{1}\)	\(m_{1}\)	\(\tau _{1}\)		\(\overline{\hat{\mu }}\)	MSE(\(\hat{\mu }\))	RAB(\(\hat{\mu }\))	AMSE	\(\overline{\check{\mu }}\)	MSE(\(\check{\mu }\))	RAB(\(\check{\mu }\))	AMSE	\(\overline{d_{1}}\)
\(\hbar\)	N	\(n_{2}\)	\(m_{2}\)	\(\tau _{2}\)	CS	\(\overline{\hat{\delta }}\)	MSE(\(\hat{\delta }\))	RAB(\(\hat{\delta }\))	ARAB	\(\overline{\check{\delta }}\)	MSE(\(\check{\delta }\))	RAB(\(\check{\delta }\))	ARAB	\(\overline{d_{2}}\)
2	40	20	12	2.0	1	1.6730	1.3543	0.6052	0.8181	1.9700	1.9392	0.7162	0.8777	4
		20	12	1.5		1.2687	1.6291	0.9871	0.6452	1.2806	1.1538	0.8549	0.6884	3
						0.9020	0.2620	0.5551		0.8620	0.3790	0.6633
						0.2882	0.0272	0.4333		0.3258	0.0390	0.5194
					2	1.6311	1.4646	0.6173	0.8457	1.9318	1.8072	0.7000	0.8168	5
						1.2757	1.5808	0.9732	0.6624	1.2724	0.9941	0.7553	0.6709	4
						0.9426	0.3096	0.6122		0.8665	0.4271	0.7028
						0.2836	0.0277	0.4467		0.3310	0.0389	0.5254
					3	1.6712	1.4729	0.6305	0.8978	1.9398	1.9019	0.7180	0.8884	4
						1.2952	1.8535	1.0302	0.6584	1.3529	1.3099	0.8947	0.6856	3
						0.8792	0.2380	0.5396		0.8182	0.3014	0.5965
						0.2931	0.0268	0.4334		0.3390	0.0405	0.5332
					4	1.7065	1.7162	0.6601	0.8627	1.9678	2.0097	0.7290	1.0619	4
						1.1531	1.5287	0.9602	0.6203	1.4402	1.9704	1.1555	0.7415	3
						0.8705	0.1812	0.4505		0.8002	0.2200	0.5208
						0.2803	0.0248	0.4102		0.3393	0.0474	0.5608
				4.0	1	1.6928	1.5273	0.6463	0.8407	1.8039	1.8936	0.7118	1.0184	9
				3.5		1.2119	1.5853	0.9712	0.6414	1.4490	1.9375	1.1282	0.7130	10
						0.8849	0.2200	0.5039		0.7927	0.2014	0.4925
						0.2899	0.0302	0.4440		0.3300	0.0413	0.5195
					2	1.6372	1.4802	0.6265	0.8358	1.8762	1.6516	0.6602	0.7989	10
						1.2821	1.6117	0.9817	0.6434	1.3216	1.2642	0.8760	0.6426	11
						0.8955	0.2238	0.5304		0.7598	0.2426	0.5389
						0.2881	0.0275	0.4350		0.3390	0.0373	0.4955
					3	1.7198	1.7149	0.6779	0.8360	1.8362	1.6565	0.6687	0.9691	9
						1.1298	1.3961	0.9223	0.6290	1.4399	1.9561	1.1345	0.7129	9
						0.8799	0.2049	0.4838		0.8258	0.2219	0.5198
						0.2869	0.0280	0.4318		0.3272	0.0420	0.5285
					4	1.7301	1.7304	0.6755	0.8908	1.9738	2.0424	0.7448	1.0624	8
						1.1796	1.6190	0.9880	0.6380	1.4244	1.9400	1.1306	0.7359	8
						0.8658	0.1864	0.4581		0.8202	0.2202	0.5164
						0.2870	0.0274	0.4304		0.3356	0.0471	0.5516
			16	2.0	1	1.7740	1.7492	0.6753	0.8599	1.9366	2.0243	0.7324	0.9353	4
			16	1.5		1.1741	1.5386	0.9603	0.6105	1.3195	1.4664	0.9845	0.6890	3
						0.8283	0.1289	0.4028		0.7822	0.2095	0.5107
						0.2914	0.0230	0.4035		0.3378	0.0408	0.5283
					2	1.7122	1.4990	0.6293	0.8688	2.0237	2.0485	0.7289	0.8426	5
						1.2894	1.7734	1.0304	0.6401	1.2392	1.0784	0.8190	0.6398	4
						0.8381	0.1753	0.4704		0.7388	0.2053	0.5075
						0.2981	0.0274	0.4305		0.3496	0.0383	0.5039
					3	1.7405	1.7201	0.6719	0.8748	1.9591	1.9860	0.7076	0.8724	4
						1.2320	1.5721	0.9839	0.6365	1.3258	1.2692	0.9021	0.6542	4
						0.8519	0.1810	0.4657		0.7372	0.1940	0.4924
						0.2930	0.0260	0.4245		0.3516	0.0403	0.5145
					4	1.7411	1.7510	0.6781	0.8804	1.9956	1.9497	0.7186	1.0211	4
						1.1861	1.5967	0.9806	0.6223	1.4485	1.9413	1.1363	0.7061	3
						0.8401	0.1489	0.4153		0.7410	0.1502	0.4378
						0.2895	0.0250	0.4150		0.3543	0.0431	0.5318
				4.0	1	1.6991	1.6230	0.6567	0.8130	1.8916	1.9040	0.7084	0.9759	10
				3.5		1.1513	1.4709	0.9615	0.6023	1.4230	1.7970	1.0925	0.6865	11
						0.8449	0.1364	0.4048		0.7798	0.1643	0.4473
						0.2815	0.0216	0.3863		0.3351	0.0384	0.4978
					2	1.7754	1.7039	0.6628	0.9496	1.9611	2.0128	0.7325	0.8514	12
						1.3530	1.9060	1.0952	0.6616	1.2904	1.1442	0.8519	0.6527	13
						0.8120	0.1598	0.4491		0.7649	0.2099	0.5166
						0.3097	0.0288	0.4393		0.3427	0.0389	0.5097
					3	1.7685	1.5754	0.6348	0.8235	1.9091	1.9200	0.7227	0.9759	11
						1.2176	1.5609	0.9753	0.6024	1.4201	1.7721	1.0651	0.6933	11
						0.8096	0.1342	0.3961		0.7712	0.1684	0.4565
						0.2996	0.0236	0.4035		0.3438	0.0433	0.5288
					4	1.8024	1.8721	0.6942	0.8770	1.9223	1.8233	0.7006	0.9365	10
						1.1156	1.4602	0.9356	0.6143	1.3787	1.7363	1.0638	0.6725	11
						0.8457	0.1513	0.4150		0.7579	0.1476	0.4264
						0.2858	0.0244	0.4123		0.3400	0.0390	0.4993
			20	–	–	1.7299	1.6335	0.6462	0.8400	1.8386	1.8867	0.7036	0.9090	–
			20			1.2046	1.5875	0.9791	0.5963	1.3425	1.5659	0.9996	0.6548	–
						0.8167	0.1175	0.3741		0.7699	0.1476	0.4262
						0.2918	0.0215	0.3858		0.3323	0.0358	0.4899
2	60	30	18	2.0	1	1.6728	1.3786	0.6020	0.8709	1.7983	1.4493	0.6256	0.6511	6
		30	18	1.5		1.3715	1.9030	1.0647	0.6416	1.2409	0.9017	0.7216	0.5879	5
						0.8187	0.1741	0.4675		0.7448	0.2192	0.5241
						0.3086	0.0280	0.4324		0.3433	0.0340	0.4804
					2	1.6123	1.2288	0.5665	0.8167	1.8619	1.4103	0.6276	0.6580	7
						1.4298	1.8067	1.0597	0.6421	1.2308	0.9283	0.7017	0.5865	5
						0.8341	0.2025	0.5068		0.7788	0.2607	0.5510
						0.3122	0.0287	0.4354		0.3377	0.0328	0.4657
					3	1.6468	1.4402	0.6099	0.8593	1.8520	1.6317	0.6668	0.7227	6
						1.3810	1.8079	1.0692	0.6380	1.2709	1.0052	0.7755	0.614	4
						0.8128	0.1641	0.4576		0.7512	0.2168	0.5194
						0.3063	0.0251	0.4155		0.3437	0.0369	0.4941
					4	1.6504	1.4739	0.6327	0.7964	1.9315	1.8738	0.7011	0.9253	5
						1.2069	1.5570	0.9844	0.5969	1.4008	1.6418	1.0536	0.6681	4
						0.8347	0.1334	0.3910		0.7475	0.1475	0.4167
						0.2838	0.0214	0.3794		0.3449	0.0379	0.5011
				4.0	1	1.6366	1.4533	0.6200	0.7896	1.8192	1.5981	0.6567	0.8564	13
				3.5		1.2730	1.5443	0.9683	0.5942	1.3970	1.6377	1.0218	0.6470	14
						0.8143	0.1381	0.4052		0.7608	0.1539	0.4328
						0.3007	0.0225	0.3832		0.3358	0.0359	0.4768
					2	1.6537	1.3527	0.6009	0.7755	1.8108	1.5108	0.6383	0.7006	15
						1.3281	1.5589	0.9806	0.6111	1.3009	1.0742	0.7997	0.5964	16
						0.8141	0.1650	0.4519		0.6963	0.1829	0.4734
						0.3087	0.0256	0.4111		0.3519	0.0345	0.4741
					3	1.6658	1.5330	0.6394	0.8148	1.8197	1.6630	0.6664	0.8904	13
						1.2249	1.5567	0.9816	0.6027	1.4373	1.7196	1.0615	0.6627	13
						0.8394	0.1481	0.4124		0.7447	0.1411	0.4215
						0.2914	0.0212	0.3776		0.3438	0.0379	0.5013
					4	1.7420	1.4577	0.6233	0.7935	2.0186	2.0407	0.7311	1.0060	11
						1.2581	1.5853	1.0158	0.5984	1.3924	1.7977	1.0800	0.6861	12
						0.7858	0.1071	0.3567		0.7473	0.1466	0.4258
						0.3072	0.0238	0.3979		0.3470	0.0390	0.5076
			24	2.0	1	1.6811	1.4593	0.6177	0.8396	1.8312	1.5713	0.6398	0.8140	6
			24	1.5		1.2960	1.7597	1.0449	0.6105	1.3955	1.5108	0.9905	0.6329	5
						0.7948	0.1153	0.3785		0.7145	0.1362	0.4124
						0.3032	0.0240	0.4011		0.3534	0.0378	0.4888
					2	1.6710	1.4174	0.6147	0.8031	1.8316	1.4141	0.6202	0.6162	7
						1.3185	1.6533	1.0194	0.6028	1.2030	0.8599	0.7410	0.5656	5
						0.7853	0.1190	0.3870		0.7197	0.1597	0.4483
						0.3065	0.0228	0.3902		0.3427	0.0309	0.4530
					3	1.6895	1.5036	0.6309	0.8307	1.8869	1.6566	0.6704	0.7610	6
						1.3103	1.6813	1.0314	0.6069	1.3089	1.1992	0.8704	0.6161	5
						0.7855	0.1161	0.3787		0.7170	0.1526	0.4427
						0.3060	0.0220	0.3865		0.3514	0.0355	0.4811
					4	1.7448	1.5816	0.6397	0.7986	1.8661	1.7059	0.6816	0.8623	6
						1.1782	1.4921	0.9625	0.5765	1.3616	1.5881	1.0120	0.6378	5
						0.7968	0.1018	0.3428		0.7430	0.1210	0.3873
						0.2941	0.0190	0.3609		0.3383	0.0341	0.4701
				4.0	1	1.7346	1.5235	0.6292	0.8181	1.8295	1.5036	0.6429	0.8263	15
				3.5		1.2895	1.6386	1.0045	0.5827	1.4068	1.6608	1.0560	0.6319	16
						0.7732	0.0908	0.3342		0.7265	0.1097	0.3752
						0.3060	0.0195	0.3629		0.3415	0.0312	0.4534
					2	1.7481	1.6436	0.6558	0.8600	1.8407	1.5361	0.6418	0.6987	18
						1.3444	1.6573	1.0315	0.6167	1.2780	1.0618	0.8052	0.5925	19
						0.7743	0.1144	0.3811		0.7111	0.1636	0.4532
						0.3151	0.0245	0.3984		0.3490	0.0333	0.4697
					3	1.8187	1.7826	0.6679	0.8806	1.8190	1.6746	0.6677	0.8485	16
						1.2442	1.6276	0.9909	0.5929	1.3730	1.5593	1.0010	0.6328	17
						0.7739	0.0921	0.3384		0.7482	0.1268	0.3988
						0.3056	0.0200	0.3743		0.3354	0.0334	0.4638
					4	1.7345	1.5845	0.6395	0.8099	1.8163	1.5654	0.6473	0.8374	15
						1.2444	1.5417	0.9765	0.5802	1.3890	1.6319	1.0282	0.6316	16
						0.7838	0.0933	0.3384		0.7303	0.1183	0.3838
						0.3018	0.0201	0.3664		0.3432	0.0341	0.4671
			30	–	–	1.7431	1.6372	0.6556	0.8263	1.8132	1.5076	0.6381	0.7707	–
			30			1.2965	1.5735	0.9866	0.5715	1.3232	1.4382	0.9650	0.6011	–
						0.7604	0.0775	0.3054		0.7347	0.1084	0.3652
						0.3068	0.0168	0.3382		0.3348	0.0287	0.4363

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