BARRETO-SOUZA, WAGNER; LEMONTE, ARTUR J.; CORDEIRO, GAUSS M.

doi:10.1590/S0001-37652013000100002

Figuras (2)
Tabelas (8)

Figure 1 Estimated pdf of the Marshall-Olkin extended exponential (M-O-EE), Weibull and gamma models for a real data set.

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Figure 2 Estimated hazard rate of the Marshall-Olkin extended exponential (M-O-EE), Weibull and gamma models for a real data set.

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TABLE I Point estimates for n=50 and different values of α.

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Method of moments α Estimates of α Estimates of λ Mean Rel. Bias Mean Rel. Bias 0.2 0.3781 0.8906 0.3438 0.8216 0.6432 0.6851 0.6 0.8682 0.4469 0.3256 0.6601 0.3201 0.2320 1.5 1.1207 -0.2528 0.4437 0.4197 -0.1607 0.1129 1.8 1.2927 -0.2818 0.6802 0.4064 -0.1872 0.1328 2.0 1.4299 -0.2850 0.8329 0.4033 -0.1935 0.1429 2.5 1.8539 -0.2584 1.2343 0.4058 -0.1884 0.1582 3.0 2.3946 -0.2018 1.7244 0.4160 -0.1680 0.1671 4.0 3.6432 -0.0892 3.0690 0.4338 -0.1324 0.1843 5.0 5.0381 0.0076 4.8922 0.4446 -0.1108 0.1971 7.0 8.2577 0.1797 11.7298 0.4591 -0.0818 0.3087 10.0 13.4647 0.3465 25.7848 0.4603 -0.0794 0.3387 20.0 28.3708 0.4185 53.4771 0.4343 -0.1315 0.2879 MSE: mean squared error; Rel. Bias: relative bias.

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TABLE II Point estimates for n=50 and different values of α.

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Method of moments α Estimates of α Estimates of λ Mean Rel. Bias Mean Rel. Bias 0.2 0.2881 0.4407 0.2331 0.6595 0.3191 0.4515 0.6 0.7899 0.3165 0.2714 0.6090 0.2180 0.1732 1.5 1.2103 -0.1932 0.3948 0.4364 -0.1272 0.0936 1.8 1.4839 -0.1756 0.5785 0.4397 -0.1206 0.1080 2.0 1.6932 -0.1534 0.6985 0.4453 -0.1095 0.1133 2.5 2.2637 -0.0945 1.0076 0.4595 -0.0811 0.1192 3.0 2.8963 -0.0346 1.3633 0.4731 -0.0539 0.1212 4.0 4.2494 0.0624 2.3092 0.4919 -0.0162 0.1262 5.0 5.6095 0.1219 3.5487 0.5003 0.0005 0.1330 7.0 8.5945 0.2278 8.6495 0.5117 0.0235 0.1461 10.0 13.3775 0.3378 15.3995 0.5182 0.0365 0.1606 20.0 32.1990 0.6099 53.5464 0.5267 0.0534 0.1878 MSE: mean squared error; Rel. Bias: relative bias.

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TABLE III Integrated relative bias squared norm.

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TABLE IV Average root mean squared error.

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TABLE V MLEs of the model parameters and log-likelihood functions.

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TABLE VI MLEs of the model parameters and log-likelihood functions.

Maximum likelihood
α	Estimates of α			Estimates of λ
α	Mean	Rel. Bias		Mean	Rel. Bias
0.2	0.2815	0.4073	0.2041	0.6324	0.2649	0.3495
0.6	0.7503	0.2505	0.4355	0.5572	0.1144	0.1953
1.5	1.8160	0.2107	0.9866	0.5329	0.0658	0.1358
1.8	2.1746	0.2081	1.1797	0.5300	0.0600	0.1278
2.0	2.4145	0.2073	1.3108	0.5285	0.0571	0.1236
2.5	3.0171	0.2068	1.6465	0.5258	0.0516	0.1156
3.0	3.6235	0.2078	1.9928	0.5239	0.0478	0.1099
4.0	4.8465	0.2116	2.7143	0.5215	0.0429	0.1021
5.0	6.0819	0.2164	3.4705	0.5199	0.0399	0.0970
7.0	8.5851	0.2264	5.0745	0.5181	0.0362	0.0905
10.0	12.4085	0.2409	7.6766	0.5166	0.0332	0.0849
20.0	25.6060	0.2803	17.6683	0.5147	0.0294	0.0771

Method of moments
α	Estimates of α			Estimates of λ
α	Mean	Rel. Bias		Mean	Rel. Bias
0.2	0.3781	0.8906	0.3438	0.8216	0.6432	0.6851
0.6	0.8682	0.4469	0.3256	0.6601	0.3201	0.2320
1.5	1.1207	-0.2528	0.4437	0.4197	-0.1607	0.1129
1.8	1.2927	-0.2818	0.6802	0.4064	-0.1872	0.1328
2.0	1.4299	-0.2850	0.8329	0.4033	-0.1935	0.1429
2.5	1.8539	-0.2584	1.2343	0.4058	-0.1884	0.1582
3.0	2.3946	-0.2018	1.7244	0.4160	-0.1680	0.1671
4.0	3.6432	-0.0892	3.0690	0.4338	-0.1324	0.1843
5.0	5.0381	0.0076	4.8922	0.4446	-0.1108	0.1971
7.0	8.2577	0.1797	11.7298	0.4591	-0.0818	0.3087
10.0	13.4647	0.3465	25.7848	0.4603	-0.0794	0.3387
20.0	28.3708	0.4185	53.4771	0.4343	-0.1315	0.2879

Maximum likelihood
α	Estimates of α			Estimates of λ
α	Mean	Rel. Bias		Mean	Rel. Bias
0.2	0.2254	0.1268	0.0893	0.5438	0.0877	0.1697
0.6	0.6470	0.0784	0.2028	0.5194	0.0388	0.1030
1.5	1.5980	0.0653	0.4625	0.5113	0.0225	0.0740
1.8	1.9158	0.0643	0.5517	0.5103	0.0205	0.0700
2.0	2.1278	0.0639	0.6119	0.5098	0.0195	0.0678
2.5	2.6588	0.0635	0.7647	0.5088	0.0176	0.0637
3.0	3.1906	0.0635	0.9207	0.5082	0.0163	0.0607
4.0	4.2570	0.0642	1.2413	0.5073	0.0146	0.0566
5.0	5.3265	0.0653	1.5720	0.5068	0.0135	0.0538
7.0	7.4736	0.0677	2.2596	0.5061	0.0122	0.0503
10.0	10.7113	0.0711	3.3452	0.5056	0.0112	0.0473
20.0	21.6129	0.0806	7.3070	0.5049	0.0098	0.0429

Method of moments
α	Estimates of α			Estimates of λ
α	Mean	Rel. Bias		Mean	Rel. Bias
0.2	0.2881	0.4407	0.2331	0.6595	0.3191	0.4515
0.6	0.7899	0.3165	0.2714	0.6090	0.2180	0.1732
1.5	1.2103	-0.1932	0.3948	0.4364	-0.1272	0.0936
1.8	1.4839	-0.1756	0.5785	0.4397	-0.1206	0.1080
2.0	1.6932	-0.1534	0.6985	0.4453	-0.1095	0.1133
2.5	2.2637	-0.0945	1.0076	0.4595	-0.0811	0.1192
3.0	2.8963	-0.0346	1.3633	0.4731	-0.0539	0.1212
4.0	4.2494	0.0624	2.3092	0.4919	-0.0162	0.1262
5.0	5.6095	0.1219	3.5487	0.5003	0.0005	0.1330
7.0	8.5945	0.2278	8.6495	0.5117	0.0235	0.1461
10.0	13.3775	0.3378	15.3995	0.5182	0.0365	0.1606
20.0	32.1990	0.6099	53.5464	0.5267	0.0534	0.1878

n	Estimates of α		Estimates of λ
n	MLE	MME	MLE	MME
50	0.2458	0.3720	0.0940	0.2480
150	0.0749	0.2818	0.0314	0.1312

n	Estimates of α		Estimates of λ
n	MLE	MME	MLE	MME
50	3.6967	8.7115	0.1341	0.2456
150	1.6107	7.3334	0.0716	0.1611

Distribution	Statistics
Distribution	W*	A*
M-O-EE	0.08525	0.58468
Weibull	0.12316	0.83717
Gamma	0.22368	1.48280

[TFN01t02] MSE: mean squared error; Rel. Bias: relative bias.

Estimates
Distribution	α	λ	γ
M-O-EE	0.4112	0.1618		-98.4665
M-O-EE	(0.2407)	(0.0691)	0.9066
Weibull		0.3169	(0.1002)	-99.4840
Weibull		(0.0751)	0.9453
Gamma		0.2517	(0.1777)	-99.8571
Gamma		(0.0618)

General results for the Marshall and Olkin's family of distributions

Resumo