The Transmuted Marshall-Olkin Extended Lomax Distribution

SILVA, RENILMA P. DA; CYSNEIROS, AUDREY HELEN M.A.; CORDEIRO, GAUSS M.; TABLADA, CLAUDIO JAVIER

doi:10.1590/0001-3765202020180777

RENILMA P. DA SILVA

Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Cidade Universitária, s/n, 50740-540 Recife, PE, Brazil

Universidade Federal de Rio Grande do Norte, Centro de Ensino Superior do Seridó, Departamento de Ciências Exatas e Aplicadas, Rua Joaquim Gregório, s/n, 59300-000 Caicó, RN, Brazil

https://orcid.org/0000-0002-0308-9077

AUDREY HELEN M.A. CYSNEIROS

Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Cidade Universitária, s/n, 50740-540 Recife, PE, Brazil

https://orcid.org/0000-0002-8920-4004

GAUSS M. CORDEIRO

Universidade Federal de Pernambuco, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Cidade Universitária, s/n, 50740-540 Recife, PE, Brazil

https://orcid.org/0000-0002-3052-6551

CLAUDIO JAVIER TABLADA

Universidade Federal da Paraíba, Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Cidade Universitária, s/n, 58051-900 João Pessoa, PB, Brazil

https://orcid.org/0000-0002-9482-2026

Correspondence to: Renilma Pereira da Silva E-mail: renilmatematica@yahoo.com.br

AUTHOR CONTRIBUTIONS

The results in this paper are part of the doctoral thesis of the first author (Renilma P. da Silva) under the supervision of the second author (Audrey H.M.A. Cysneiros). All authors developed jointly the paper and contributed to the discussion and text review. Renilma P. da Silva and Audrey H.M.A. Cysneiros obtained some structural properties of the new model and performed the simulation study. Gauss M. Cordeiro and Claudio J. Tablada derived useful mathematical expansions for some statistical quantities.

$α$	$β$	$γ$	$λ$	Model	Reference
$α$	$β$	$γ$	0	MOELx ( $α, β, γ$ )	^{Ghitany et al. (2007)}B13 GHITANY ME, AL-AWADHI FA & ALKHALFAN LA. 2007. Marshall-Olkin Extended Lomax Distribution and Its Application to Censored Data. Commun Stat Theory Methods 36: 1855-1866.
1	$β$	$γ$	$λ$	TLx ( $β, γ, λ$ )	^{Ashour & Eltehiwy (2013)}B5 ASHOUR SK & ELTEHIWY MA. 2013. Transmuted Lomax distribution. Am J Math Stat 1: 121-127.
1	$β$	$γ$	0	Lx ( $β, γ$ )	^{Lomax (1954)}B20 LOMAX KS. 1954. Business Failures: Another Example of the Analysis of Failure Data. J Am Stat Assoc 49: 847-852.

Parameter	Method	$N$	mean	variance	skewness	kurtosis
$α$ =0.75	Truncation	5	0.7356	0.7407	5.1369	134.01
$β$ =0.5			10	0.6960	0.6916	5.2047
$γ$ =4.5			20	0.6963	0.6923	5.2040
$λ$ =-0.75	Num. int.		0.6963	0.6923	5.2040	138.07
	Monte Carlo		0.6949	0.7069	5.7817	99.360
$α$ =1.5	Truncation	5	0.3861	0.1688	3.8940	55.140
$β$ =1.0			10	0.3933	0.1715	3.8878
$γ$ =5.0			20	0.3934	0.1715	3.8877
$λ$ =-0.5	Num. int.		0.3934	0.1715	3.8877	54.930
	Monte Carlo		0.3927	0.1735	4.3059	56.020
$α$ =3.0	Truncation	5	0.1621	0.0344	3.1759	32.507
$β$ =2.0			10	0.2060	0.0402	3.0934
$γ$ =5.5			20	0.2174	0.0420	3.0889
$λ$ =-0.25	Num. int.		0.2178	0.0421	3.0887	31.238
	Monte Carlo		0.2175	0.0424	3.3848	35.354
$α$ =5.0	Truncation	5	0.0956	0.0117	2.8857	24.900
$β$ =3.0			10	0.1239	0.0137	2.7908
$γ$ =6.0			20	0.1314	0.0143	2.7898
$λ$ =0.25	Num. int.		0.1304	0.0142	2.7891	23.886
	Monte Carlo		0.1302	0.0142	3.0287	27.900
$α$ =7.0	Truncation	5	0.0587	0.0043	2.4756	17.285
$β$ =4.0			10	0.0820	0.0051	2.3795
$γ$ =7.0			20	0.0888	0.0053	2.4155
$λ$ =0.5	Num. int.		0.0828	0.0048	2.3729	17.031
	Monte Carlo		0.0827	0.0048	2.5454	20.037

			relative bias				MSE
$λ$	$α$	$n$	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$
-0.5	0.25	100	0.954	0.975	0.057	-0.086	0.687	0.932	0.012	0.142
		200	0.549	0.696	0.045	-0.049	0.144	0.337	0.007	0.124
		250	0.471	0.632	0.040	-0.049	0.111	0.301	0.006	0.117
	0.5	100	0.742	0.884	0.021	-0.158	2.155	1.025	0.007	0.153
		200	0.403	0.695	0.020	-0.073	0.521	0.290	0.004	0.139
		250	0.348	0.671	0.020	-0.059	0.225	0.279	0.003	0.133
0.25	0.25	100	1.245	1.632	$0.098$	-0.349	2.260	3.752	0.021	0.151
		200	0.495	0.751	$0.063$	-0.312	0.215	0.432	0.013	0.126
		250	0.382	0.570	$0.049$	-0.272	0.117	0.238	0.010	0.109
	0.5	100	0.802	1.122	$0.043$	-0.110	3.145	1.638	0.013	0.122
		200	0.376	0.528	$0.001$	$0.062$	1.097	0.452	0.008	0.119
		250	0.277	0.429	-0.011	$0.076$	0.301	0.144	0.007	0.122
0.5	0.25	100	0.931	1.131	0.144	-0.292	1.436	1.867	0.028	0.136
		200	0.379	0.462	0.087	-0.202	0.227	0.297	0.018	0.105
		250	0.308	0.385	0.062	-0.154	0.153	0.221	0.015	0.096
	0.5	100	0.772	1.057	0.101	-0.324	4.467	2.560	0.019	0.121
		200	0.237	0.348	0.040	-0.222	0.609	0.288	0.011	0.101
		250	0.144	0.249	0.022	-0.196	0.379	0.122	0.009	0.098

[1] Correspondence to: Renilma Pereira da Silva E-mail: renilmatematica@yahoo.com.br

	MLE
Distribution	$\hat{α}$	$\hat{β}$	$\hat{γ}$	$\hat{λ}$
TMOELx	2,087	0,775	1,062	-0,99
		(0.497)	(0.247)	(0.141)
MOELx	118,067	7,962	1,227	-
		(179.861)	(10.360)	(0.095)
TLx	-	0,067	1,679	0,189
		-	(0.040)	(0.467)
Lx	-	0,083	1,610	-
		-	(0.031)	(0.392)

Distribution	AIC	BIC	HQIC	W*	A*
TMOELx	928,168	939,576	932,803	0,256	1,528
MOELx	933,116	941,672	936,593	0,334	2,029
TLx	935,166	943,722	938,642	0,361	2,213
Lx	933,298	939,002	935,616	0,360	2,207

Distribution	W*	A*
TMOELx	0,034	0,249
TMOFr	0,116	0,813
EsLx	0,698	4,487
ELx	0,051	0,370
BLx	0,401	2,662
KwLx	0,085	0,632

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The Transmuted Marshall-Olkin Extended Lomax Distribution

Abstract