The Exponentiated Power Generalized Weibull: Properties and Applications

PEÑA-RAMÍREZ, FERNANDO A.; GUERRA, RENATA R.; CORDEIRO, GAUSS M.; MARINHO, PEDRO R.D.

doi:10.1590/0001-3765201820170423

FERNANDO A. PEÑA-RAMÍREZ

Departamento de Estatística, Universidade Federal de Santa Maria, Cidade Universitária, Av. Roraima, 1000, 97105-900 Santa Maria, RS, Brazil

RENATA R. GUERRA

Departamento de Estatística, Universidade Federal de Santa Maria, Cidade Universitária, Av. Roraima, 1000, 97105-900 Santa Maria, RS, Brazil

GAUSS M. CORDEIRO

Departamento de Estatística, Universidade Federal de Pernambuco, Cidade Universitária,Av. Prof. Moraes Rego, 1235, 50740-540 Recife, PE, Brazil

PEDRO R.D. MARINHO

Departamento de Estatística, Universidade Federal da Paraíba, Cidade Universitária,Conj. Pres. Castelo Branco III, s/n, 58051-900 João Pessoa, PB, Brazil

Correspondence to: Fernando A. Peña-Ramírez E-mail: fernando.p.ramirez@ufsm.br

α	β	λ	γ	Distribution
1	1	-	1	Exponential
1	-	-	1	Exponentiated exponential
1	1	-	2	Rayleigh
1	-	-	2	Burr type X
1	1	-	-	Weibull
1	-	-	-	Exponentiated Weibull
-	1	-	1	Nadarajah-Haghighi
-	-	-	1	Exponentiated Nadarajah-Haghighi
-	1	-	-	Power generalized Weibull

$β$	$E (T)$	$E (T^{2})$	$E (T^{3})$	$E (T^{4})$	$E (T^{5})$	$E (T^{6})$
0.5	0.2989	0.6249	2.7433	19.2657	191.2419	2498.7853
0.8	0.4488	0.9827	4.3653	30.7679	305.7880	3997.1257
1.0	0.5397	1.2149	5.4372	38.4129	382.0702	4995.6341
2.0	0.9176	2.3100	10.6904	76.3690	762.5177	9983.6001
3.0	1.2116	3.3145	15.7832	113.8992	1141.4011	14964.0482
4.0	1.4537	4.2472	20.7336	151.0300	1518.7741	19937.1200
0.5	0.3162	0.3259	0.5425	1.2207	3.4227	11.4158
0.8	0.4515	0.4986	0.8507	1.9344	5.4503	18.2210
1.0	0.5278	0.6062	1.0498	2.4030	6.7917	22.7400
2.0	0.8109	1.0755	1.9810	4.6677	13.3819	45.1295
3.0	1.0030	1.4633	2.8250	6.8184	19.7945	67.1961
4.0	1.1481	1.7964	3.6014	8.8727	26.0480	88.9632
0.5	0.3450	0.2989	0.3824	0.6249	1.2207	2.7433
0.8	0.4789	0.4488	0.5924	0.9827	1.9344	4.3653
1.0	0.5517	0.5397	0.7257	1.2149	2.4030	5.4372
2.0	0.8066	0.9176	1.3275	2.3100	4.6677	10.6904
3.0	0.9688	1.2116	1.8477	3.3145	6.8184	15.7832
4.0	1.0868	1.4537	2.3092	4.2472	8.8727	20.7336
0.5	0.4818	0.3450	0.3012	0.2989	0.3259	0.3824
0.8	0.6092	0.4789	0.4391	0.4488	0.4986	0.5924
1.0	0.6702	0.5517	0.5194	0.5397	0.6062	0.7257
2.0	0.8518	0.8066	0.8308	0.9176	1.0755	1.3275
3.0	0.9488	0.9688	1.0538	1.2116	1.4633	1.8477
4.0	1.0129	1.0868	1.2278	1.4537	1.7964	2.3092
0.5	0.5773	0.4186	0.3450	0.3099	0.2968	0.2989
0.8	0.6929	0.5504	0.4789	0.4459	0.4377	0.4488
1.0	0.7445	0.6168	0.5517	0.5234	0.5210	0.5397
2.0	0.8868	0.8277	0.8066	0.8158	0.8527	0.9176
3.0	0.9571	0.9482	0.9688	1.0181	1.0976	1.2116
4.0	1.0018	1.0306	1.0868	1.1725	1.2925	1.4537
0.5	0.6440	0.4818	0.3950	0.3450	0.3162	0.3012
0.8	0.7478	0.6092	0.5276	0.4789	0.4515	0.4391
1.0	0.7922	0.6702	0.5961	0.5517	0.5278	0.5194
2.0	0.9091	0.8518	0.8193	0.8066	0.8109	0.8308
3.0	0.9644	0.9488	0.9507	0.9688	1.0030	1.0538
4.0	0.9987	1.0129	1.0420	1.0868	1.1481	1.2278

Mean estimates								RMSEs
n	α	β	λ	γ	α	$\hat{β}$	$\hat{λ}$	$\hat{γ}$	$\hat{α}$	$\hat{β}$	$\hat{λ}$	$\hat{γ}$
100	0.3	4.0	3.0	1.6	0.355	4.041	3.061	2.140	0.313	1.988	2.084	1.376
	1.7	0.8	0.1	0.2	1.670	0.795	0.105	0.253	0.833	0.434	0.109	0.117
	3.0	2.0	5.0	0.6	3.848	2.212	5.512	0.627	2.027	1.104	3.112	0.157
	3.5	0.9	0.2	0.1	2.729	0.781	0.284	0.166	1.674	0.493	0.276	0.128
	7.0	1.5	5.0	0.2	7.296	1.643	5.215	0.199	1.717	0.504	1.644	0.027
	7.5	1.3	4.0	0.5	8.034	1.436	4.786	0.510	2.488	0.589	2.386	0.100
300	0.3	4.0	3.0	1.6	0.316	4.035	2.990	1.843	0.145	1.515	1.534	0.807
	1.7	0.8	0.1	0.2	1.630	0.762	0.094	0.235	0.657	0.271	0.052	0.086
	3.0	2.0	5.0	0.6	3.454	2.096	5.083	0.603	1.277	0.573	2.189	0.086
	3.5	0.9	0.2	0.1	3.155	0.858	0.233	0.117	1.063	0.278	0.111	0.050
	7.0	1.5	5.0	0.2	7.093	1.556	5.121	0.199	1.130	0.285	1.141	0.018
	7.5	1.3	4.0	0.5	7.774	1.337	4.431	0.507	1.753	0.303	1.645	0.066
500	0.3	4.0	3.0	1.6	0.308	4.005	2.990	1.766	0.105	1.296	1.344	0.613
	1.7	0.8	0.1	0.2	1.639	0.766	0.093	0.226	0.572	0.226	0.041	0.069
	3.0	2.0	5.0	0.6	3.322	2.062	5.030	0.600	0.991	0.429	1.902	0.068
	3.5	0.9	0.2	0.1	3.301	0.883	0.219	0.107	0.789	0.204	0.073	0.025
	7.0	1.5	5.0	0.2	7.090	1.530	5.090	0.200	0.938	0.218	0.964	0.015
	7.5	1.3	4.0	0.5	7.724	1.323	4.272	0.504	1.462	0.231	1.327	0.053
800	0.3	4.0	3.0	1.6	0.304	4.013	2.994	1.718	0.083	1.134	1.167	0.498
	1.7	0.8	0.1	0.2	1.640	0.770	0.095	0.220	0.509	0.192	0.033	0.059
	3.0	2.0	5.0	0.6	3.242	2.044	4.993	0.599	0.812	0.336	1.656	0.056
	3.5	0.9	0.2	0.1	3.383	0.892	0.211	0.103	0.603	0.160	0.051	0.016
	7.0	1.5	5.0	0.2	7.063	1.518	5.061	0.200	0.789	0.171	0.801	0.012
	7.5	1.3	4.0	0.5	7.620	1.318	4.187	0.502	1.215	0.182	1.104	0.043

Statistics	Real data sets
Statistics	Remission times data	Stress-rupture data
Mean	9.3656	1.0248
Median	6.3950	0.8000
Mode	5.0000	0.5000
Variance	110.4250	1.2529
Skewness	3.2865	3.0017
Kurtosis	15.4831	13.7089
Maximum	79.0500	7.8900
Minimum	0.0800	0.0100
n	128	101

Distributions	Estimates
EPGW( $α, β, λ, γ$ )	0.2076	0.4062	0.0047	3.1008
	(0.0347)	(0.0547)	(0.0025)	(0.2866)
Kw-W( $a, b, c, β$ )	3.8071	1.7364	0.5144	0.2904
	(1.3951)	(0.8486)	(0.1157)	(0.1435)
BW( $a, b, α, λ$ )	6.4498	8.3256	0.3909	25.4616
	(2.7464)	(4.3776)	(0.0859)	(18.7415)
BFr( $a, b, λ, σ$ )	4.1513	8.8050	0.3087	8.8088
	(1.4516)	(2.6397)	(0.0460)	(4.0961)
PGW( $α, λ, γ$ )	0.4253	0.1364	1.5564
	(0.0996)	(0.0359)	(0.2212)
MONH( $λ, α, β$ )	1.1844	0.5025	6.3566
	(0.5670)	(0.0480)	(3.1274)
EW( $β, λ, γ$ )	0.4854	0.5421	3.9736
	(0.1821)	(0.0619)	(1.0804)
MW( $α, β, λ$ )	0.0030	0.1979	2.2188
	(0.0009)	(0.0063)	(0.6607)
ENH( $α, β, λ$ )	0.6003	0.4002	1.7880
	(0.0841)	(0.1614)	(0.3419)
NH( $α, λ$ )	0.9134	0.1236
	(0.1475)	(0.0344)
Chen( $β, λ$ )	0.1106	0.3538
	(0.0152)	(0.0123)
Weibull( $α, λ$ )	9.5470	1.0490
	(0.8499)	(0.0676)
FW( $α, β$ )	0.0325	2.1553
	(0.0026)	(0.2490)

Distributions	Estimates
EPGW( $α, β, λ, γ$ )	0.1349	0.1022	0.0415	6.6681
	(0.0171)	(0.0104)	(0.0154)	(0.0136)
Kw-W( $a, b, c, β$ )	0.7029	0.2175	1.0118	4.3625
	(0.1620)	(0.1038)	(0.0027)	(2.1072)
BW( $a, b, α, λ$ )	0.7482	0.2305	1.1275	0.3245
	(0.1608)	0.0338)	(0.0858)	(0.0856)
BFr( $a, b, λ, σ$ )	0.3984	5.0048	0.4208	(9.1684)
	(0.1702)	(1.4399)	(0.0534)	(4.3072)
PGW( $α, λ, γ$ )	1.2659	0.7182	0.8696
	(0.4483)	(0.3485)	(0.1039)
MONH( $λ, α, β$ )	0.0146	17.7252	0.2122
	(0.0067)	(7.7187)	(0.0618)
EW( $β, λ, γ$ )	0.8488	1.0419	0.8171
	(0.2981)	(0.2511)	(0.3157)
MW( $α, β, λ$ )	0.0027	0.2259	7.0190
	(0.0008)	(0.0076)	(1.5244)
ENH( $α, β, λ$ )	1.0732	0.7762	0.8426
	(0.2760)	(0.3582)	(0.1238)
NH( $α, λ$ )	0.8898	1.1810
	(0.1853)	(0.4270)
Chen( $β, λ$ )	0.5410	0.5303
	(0.0585)	(0.0321)
Weibull( $α, λ$ )	0.9919	0.9259
	(0.1121)	(0.0726)
FW( $α, β$ )	0.3287	0.0838
	(0.0246)	(0.0133)

Distributions	Statistics
Distributions	*W $^{}$**	*A $^{}$**	KS
EPGW	0.0166	0.1148	0.0384
Kw-W	0.0408	0.2744	0.0447
BW	0.0484	0.3313	0.0543
BFr	0.2370	1.5485	0.0987
PGW	0.0352	0.2345	0.0431
MONH	0.0832	0.4864	0.0641
EW	0.0458	0.3163	0.0483
MW	0.3636	2.1554	0.1067
ENH	0.0413	0.2781	0.0422
NH	0.0997	0.6017	0.0927
Chen	0.4430	2.6114	0.1411
Weibull	0.1318	0.7890	0.0695
FW	1.4134	7.8093	0.2085

Distributions	Statistics
Distributions	*W $^{}$**	*A $^{}$**	KS
EPGW	0.0722	0.4672	0.0699
Kw-W	0.1400	0.8478	0.1017
BW	0.2753	1.5190	0.1005
BFr	0.7116	3.8276	0.1914
PGW	0.1730	0.9930	0.0833
MONH	1.1054	5.9604	0.3068
EW	0.1686	0.9736	0.0875
MW	0.0980	0.7596	0.1292
ENH	0.1670	0.9667	0.0837
NH	0.2053	1.1434	0.0819
Chen	0.1207	0.8756	0.0973
Weibull	0.1987	1.1115	0.0900
FW	1.1130	5.9971	0.3054

\begin{matrix} h (t) = & α β λ γ t^{γ - 1} (1 + λ t^{γ})^{α - 1} \\ \times \frac{exp {1 - (1 + λ t^{γ})^{α}} {[1 - exp {1 - (1 + λ t^{γ})^{α}}]}^{β - 1}}{1 - [1 - exp {1 - (1 + λ t^{γ})^{α}}]^{β}} . \end{matrix}

\begin{matrix} ℓ (𝛉) & = & n + n log (α β λ γ) + (γ - 1) \sum_{i = 1}^{n} log (t_{i}) - \sum_{i = 1}^{n} {(1 + λ t_{i}^{γ})}^{α} \\ + & (α - 1) \sum_{i = 1}^{n} log (1 + λ t_{i}^{γ}) + (β - 1) \sum_{i = 1}^{n} log [1 - e^{1 - {(1 + λ t_{i}^{γ})}^{α}}] . \end{matrix}

\begin{matrix} ℓ (𝛉_{𝐩}) = & n & log (n) + n log (α λ γ) + (γ - 1) \sum_{i = 1}^{n} log (t_{i}) - \sum_{i = 1}^{n} {(1 + λ t_{i}^{γ})}^{α} \\ + & (α - 1) \sum_{i = 1}^{n} log (1 + λ t_{i}^{γ}) - \sum_{i = 1}^{n} log [1 - e^{1 - {(1 + λ t_{i}^{γ})}^{α}}] \\ - & n log {- \sum_{i = 1}^{n} log [1 - e^{1 - {(1 + λ t_{i}^{γ})}^{α}}]} . \end{matrix}

[1] Correspondence to: Fernando A. Peña-Ramírez E-mail: fernando.p.ramirez@ufsm.br

Brasil

Brasil

The Exponentiated Power Generalized Weibull: Properties and Applications

Abstract