The Generalized Odd Lindley-G Family: Properties and Applications

AFIFY, AHMED Z.; CORDEIRO, GAUSS M.; MAED, MOHAMED E.; ALIZADEH, MORAD; AL-MOFLEH, HAZEM; NOFAL, ZOHDY M.

doi:10.1590/0001-3765201920180040

AHMED Z. AFIFY

Department of Statistics, Mathematics and Insurance, Benha University, Benha, 13518 Egypt

GAUSS M. CORDEIRO

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

MOHAMED E. MAED

Department of Statistics, Mathematics and Insurance, Zagazig University, Zagazig, 44519 Egypt

Department of Mathematics, Faculty of Science and Arts, Khulais, University of Jeddah, Jeddah, 21959 Saudi Arabia

MORAD ALIZADEH

Department of Statistics, Faculty of Sciences Gulf University, Bushehr, 75169, Iran

HAZEM AL-MOFLEH

Department of Mathematics, Tafila Technical University, Tafila, 66110, Jordan

http://orcid.org/0000-0003-3430-2464

ZOHDY M. NOFAL

Department of Statistics, Mathematics and Insurance, Benha University, Benha, 13518 Egypt

Correspondence to: Hazem Al-Mofleh E-mail: almof1hm@cmich.edu ORCid: https://orcid.org/0000-0003-3430-2464

AUTHOR CONTRIBUTIONS

The first author introduced the idea and obtained main properties of the family and the special sub-models and applications. The second author checked all properties, improve the presentation of the paper and edit it. The third author helped in developing some properties and revision of the paper. Some important mathematical properties are developed by the fourth author. The Monte Carlo simulations were performed by the fifth author Monte Carlo simulations. The last author wrote the introduction section and worked on the application section. All authors together refined the manuscript with consultation to each other.

n	$\hat{α}$	$M S E (\hat{α})$	λ	$\hat{λ}$	$MSE (\hat{λ})$	a	$\hat{a}$	$MSE (\hat{a})$	b	$\hat{b}$	$MSE (\hat{b})$
350	2.47785	3.57641	0.5	0.50116	0.00037	0.7	0.76066	0.12152	0.9	0.91553	0.03548
550	2.26562	1.43760		0.50075	0.00026		0.73673	0.07164		0.90973	0.02123
750	2.19027	0.82444		0.49994	0.00018		0.72749	0.05207		0.90729	0.01567
350	2.48698	3.04369		0.50097	0.00037		0.76611	0.12216	3.0	3.04001	0.38174
550	2.35512	1.77756		0.50053	0.00024		0.75840	0.07948		2.99564	0.23581
750	2.18049	0.77913		0.50048	0.00018		0.72470	0.05077		3.02977	0.17295
350	2.56685	3.99778		0.50171	0.00039		0.78214	0.13450	5.0	5.02180	1.10152
550	2.29636	1.34325		0.50041	0.00023		0.74661	0.07026		5.01681	0.61638
750	2.22297	0.79798		0.50049	0.00017		0.73822	0.05116		5.00113	0.45527
350	2.52108	4.49541		0.50067	0.00040	2.5	2.57087	0.27583	0.9	0.91488	0.03812
550	2.28265	1.34818		0.50071	0.00024		2.54553	0.15601		0.90647	0.02071
750	2.16575	0.69721		0.50030	0.00018		2.52146	0.10832		0.90822	0.01519
350	2.60060	5.04284		0.50075	0.00038		2.59315	0.28616	3.0	3.01851	0.38213
550	2.32758	2.52439		0.50068	0.00024		2.55328	0.16813		3.01762	0.24144
750	2.18993	0.76092		0.50050	0.00018		2.53138	0.10957		3.01966	0.16518
350	2.49875	4.91899		0.50165	0.00036		2.56453	0.26356	5.0	5.07914	1.03591
550	2.28231	1.38029		0.50047	0.00025		2.54382	0.16013		5.04115	0.65936
750	2.20620	0.74794		0.50006	0.00018		2.53896	0.11032		5.01394	0.45806
350	2.50010	5.20868		0.50130	0.00039	4.3	4.37869	0.18204	0.9	0.91435	0.03498
550	2.27733	1.24864		0.50083	0.00025		4.35394	0.10463		0.90657	0.02101
750	2.18215	0.68739		0.50072	0.00018		4.34226	0.07059		0.90677	0.01501
350	2.63310	4.44811		0.50226	0.00039		4.41352	0.20697	3.0	3.00294	0.38661
550	2.32132	1.64614		0.50071	0.00024		4.36402	0.11117		3.01137	0.23855
750	2.22868	0.84644		0.50041	0.00018		4.35037	0.07886		3.00408	0.16942
350	2.44902	3.50202		0.50017	0.00040		4.37362	0.17580	5.0	5.11283	1.08059
550	2.32181	1.41633		0.50047	0.00024		4.36786	0.11115		5.00644	0.63025
750	2.17571	0.71741		0.50056	0.00018		4.33550	0.07252		5.04160	0.46412
350	2.30486	2.08277	2.0	2.00368	0.00750	0.7	0.74786	0.10516	0.9	0.93034	0.05804
550	2.15112	0.67356		2.00226	0.00470		0.72423	0.05835		0.91879	0.03046
750	2.10368	0.39468		2.00078	0.00340		0.71692	0.04031		0.91497	0.02208
350	2.25492	1.33882		2.00225	0.00723		0.73784	0.09783	3.0	3.12346	0.71101
550	2.13798	0.67363		1.99995	0.00435		0.71945	0.05852		3.07807	0.35595
750	2.11914	0.43386		2.00001	0.00356		0.72139	0.04315		3.04201	0.25828
350	2.28741	1.35652		2.00506	0.00747		0.74795	0.09982	5.0	5.16168	1.75064
550	2.19248	0.73292		2.00276	0.00470		0.73579	0.06384		5.07695	1.01064
750	2.12997	0.44600		2.00306	0.00358		0.72454	0.04370		5.05948	0.70877
350	2.26410	1.52295		2.00196	0.00783	2.5	2.54455	0.11435	0.9	0.93847	0.05952
550	2.17711	0.73567		2.00191	0.00462		2.53418	0.06981		0.91443	0.03068
750	2.10840	0.41262		2.00299	0.00354		2.52255	0.04742		0.91483	0.02286
350	2.27206	1.48501		2.00657	0.00725		2.55000	0.11461	3.0	3.12107	0.63362
550	2.14655	0.63521		2.00258	0.00465		2.52724	0.06762		3.07540	0.36154
750	2.10247	0.43702		2.00069	0.00343		2.51833	0.04785		3.05464	0.25187
350	2.29692	1.57601		2.00340	0.00720		2.55633	0.11672	5.0	5.18189	1.90371
550	2.13263	0.68456		2.00081	0.00445		2.51781	0.06929		5.15111	1.09036
750	2.09321	0.41101		2.00179	0.00340		2.51647	0.04651		5.09579	0.68370
350	2.32903	1.61028		2.00543	0.00762	4.3	4.45296	0.12749	0.9	0.92361	0.05614
550	2.15238	0.66621		2.00276	0.00471		4.38676	0.05886		0.92085	0.03299
750	2.09719	0.39013		2.00076	0.00330		4.35575	0.03260		0.91460	0.02165
350	2.31347	1.50773		2.00079	0.00753		4.44388	0.12973	3.0	3.07847	0.62103
550	2.13992	0.67529		2.00325	0.00451		4.39256	0.06239		3.08653	0.37296
750	2.11888	0.45106		1.99907	0.00336		4.36968	0.03603		3.04865	0.25858
350	2.24846	1.46959		2.00295	0.00746		4.44081	0.12498	5.0	5.19737	1.70506
550	2.15252	0.63939		2.00190	0.00469		4.38198	0.05560		5.11136	1.03083
750	2.07391	0.38562		2.00203	0.00346		4.36451	0.03638		5.12284	0.69942

Model	AIC	CAIC	HQIC	BIC	W^*	A^*
GOLiLx	175.045	176.187	177.487	181.800	0.1020	0.7305
GOLiLL	175.945	177.078	178.387	182.700	0.1173	0.8197
KwTLL	184.116	185.880	187.169	192.560	0.2029	1.3256
KwLL	182.470	183.613	184.913	189.226	0.2072	1.3504
TLL	$183.413$	$184.080$	$185.245$	$188.480$	$0.2161$	$1.3995$
McLL	$190.374$	$192.139$	$193.428$	$198.819$	$0.2895$	$1.8181$
BLL	$189.769$	$190.912$	$192.212$	$196.525$	$0.3068$	$1.9212$
GTLL	$211.936$	$213.701$	$214.990$	$220.381$	$0.5591$	$3.3694$
LiLL	$228.936$	$229.602$	$230.768$	$234.002$	$0.7117$	$4.4281$

Model	$λ$	$α$	$β$
TLL	0.0007	6.2257	4.8416
	(0.8200)	(0.9951)	(0.6540)
LiLL	0.0557	1.6560	27.6744
	(0.0220)	(0.0570)	(9.4810)
	$α$	$λ$	$a$	$b$
GOLiLx	73.8241	0.0692	0.4833	2.8316
	(0.0009)	(0.0500)	(1.7480 x 10⁶)	(0.2650)
GOLiLL	35.4255	0.0670	0.4230	2.5184
	(0.0011)	(0.0490)	(5.0250 x 10⁶)	(0.2580)
	$α$	$β$	$a$	$b$
KwLL	0.5882	0.7449	28.9123	77.0219
	(0.8820)	(0.1010)	(25.7920)	(61.3480)
BLL	4.9462	0.7316	25.3193	22.3871
	(1.2120)	(0.1210)	(18.7250)	(14.3040)
	$α$	$β$	$a$	$b$	λ
KwTLL	0.1739	0.6619	56.2427	118.6118	0.0006
	(0.1300)	(0.1060)	(44.7200)	(100.2550)	(1.4020)
GTLL	0.8271	2.0222	34.4721	0.7329	4.2209 x 10⁶
	(0.409)	(0.2030)	(31.1840)	(18.7070)	(0.0460)
	$α$	$β$	$a$	$b$	$c$
McLL	17.3407	0.6128	32.4696	49.3263	0.6020
	(0.2880)	(19.9640)	(28.4240)	(10.0180)	(1.4040)

Model	AIC	CAIC	HQIC	BIC	W^*	A^*
GOLiBXII	829.713	830.038	834.348	841.121	0.04336	0.28618
WBXII	829.812	830.138	834.448	841.221	0.04976	0.32650
KwEBXII	831.651	832.143	837.445	845.911	0.04817	0.31966
McW	831.680	832.172	837.474	845.940	0.05037	0.32985
BBXII	832.297	832.789	838.091	846.557	0.04116	0.29716
OLiBXII	832.793	832.987	836.270	841.350	0.11701	0.70190
BEBXII	841.268	841.760	847.062	855.528	0.13369	0.90004
LiBXII	868.200	868.394	871.677	876.756	0.37270	2.71926
BXII	910.950	911.046	913.267	916.654	0.69396	5.37094

Model	a	b
BXII	2.3354	0.2337
	(0.3540)	(0.0400)
	$λ$	a	b
OLiBXII	0.2810	0.8766	0.9669
	(0.1640)	(0.5520)	(0.7690)
LiBXII	0.0472	1.7448	12.7645
	(0.2880)	(14.0960)	(0.0470)
	$α$	$λ$	a	b
GOLiBXII	0.7684	0.4170	1.7800	0.3712
	(0.0013)	(0.0760)	(0.0037)	(0.0340)
WBXII	0.7890	0.2008	6.7391	2.4552
	(0.4018)	(0.3120)	(43.9190)
	$α$	$β$	$a$	b	$c$
McW	0.1192	0.5582	4.0633	2.6036	0.0393
	(0.1090)	(0.1780)	(2.1110)	(2.4520)	(0.2020)
	$θ$	$β$	$a$	b	c
KwEBXII	0.8389	3.0825	2.7800	67.6361	0.3383
	(1.7230)	(49.3530)	(44.5100)	(104.7280)	(0.3850)
BBXII	0.8665	6.3741	1.0891	1.3905	1.5728
	(1.0170)	(1.5820)	(0.4510)	(2.4050)	(0.4410)
BEBXII	1.3064	1.7799	22.1864	20.2769	0.2245
	(1.0790)	(1.0760)	(21.9560)	(17.2960)	(0.1440)

[1] Correspondence to: Hazem Al-Mofleh E-mail: almof1hm@cmich.edu ORCid: https://orcid.org/0000-0003-3430-2464

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The Generalized Odd Lindley-G Family: Properties and Applications