The Lindley Weibull Distribution: properties and applications

CORDEIRO, GAUSS M.; AFIFY, AHMED Z.; YOUSOF, HAITHAM M.; CAKMAKYAPAN, SELEN; OZEL, GAMZE

doi:10.1590/0001-3765201820170635

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Mathematical Sciences • An. Acad. Bras. Ciênc. 90 (3) • Jul-Sep 2018 • https://doi.org/10.1590/0001-3765201820170635 copy

The Lindley Weibull Distribution: properties and applications

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

We introduce a new three-parameter lifetime model called the Lindley Weibull distribution, which accommodates unimodal and bathtub, and a broad variety of monotone failure rates. We provide a comprehensive account of some of its mathematical properties including ordinary and incomplete moments, quantile and generating functions and order statistics. The new density function can be expressed as a linear combination of exponentiated Weibull densities. The maximum likelihood method is used to estimate the model parameters. We present simulation results to assess the performance of the maximum likelihood estimation. We prove empirically the importance and flexibility of the new distribution in modeling two data sets.

Key words
Lindley G-Family; maximum likelihood; moments; order statistics

$β$	$θ$	Mean	Variance	Skewness	Kurtosis
0.75	0.75	3.0785	12.3301	2.4522	12.6252
	1.5	1.6000	9.7659	5.3718	56.6381
	3.5	0.2358	0.2465	5.9361	69.1826
	5	0.1067	0.0526	6.1391	74.213
	10	0.0236	0.0027	6.4156	81.6319
1.5	0.75	1.5102	0.7978	0.7857	3.6215
	1.5	0.8726	0.3017	0.8783	3.7978
	3.5	0.4496	0.0883	0.9856	4.0844
	5	0.3430	0.0525	1.0168	4.1850
	10	0.2063	0.0194	1.0521	4.3112
2.5	0.75	1.2230	0.2126	0.1393	2.7452
	1.5	0.8751	0.1230	0.2156	2.7272
	3.5	0.5853	0.0600	0.2990	2.7767
	5	0.4972	0.0441	0.3216	2.8015
	10	0.3661	0.0243	0.3459	2.8351
5	0.75	1.0836	0.0486	-0.4485	3.1601
	1.5	0.9144	0.0390	-0.3759	2.9957
	3.5	0.7464	0.0281	-0.3016	2.9034
	5	0.6876	0.0242	-0.2829	2.8902
	10	0.5898	0.0181	-0.2636	2.8817
10	0.75	1.0350	0.0123	-0.8301	4.1053
	1.5	0.9501	0.0117	-0.7547	3.8264
	3.5	0.8579	0.0103	-0.6815	3.6441
	5	0.8234	0.0096	-0.6639	3.6105
	10	0.7626	0.0083	-0.6461	3.5816

			n
			30						50
Parameters			Mean Values			MSEs			Mean Values			MSEs
$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$
0.5	0.2	2.5	0.50991	0.19792	2.62564	0.00501	0.00030	0.17232	0.50649	0.19867	2.57862	0.00300	0.00020	0.10098
0.5	0.2	4.0	0.51217	0.19906	4.18515	0.00493	0.00013	0.51064	0.50734	0.19860	4.14959	0.00286	0.00008	0.26106
0.5	2.5	2.5	0.51175	2.47809	2.62782	0.00521	0.05289	0.19608	0.50506	2.48636	2.56986	0.00279	0.02888	0.09129
0.5	2.5	4.0	0.50879	2.48038	4.19961	0.00467	0.01829	0.43254	0.50495	2.48680	4.12371	0.00303	0.01091	0.23466
0.5	0.2	0.3	0.51061	0.24238	0.31474	0.00497	0.05728	0.00262	0.50846	0.21737	0.30899	0.00270	0.02213	0.00129
1.6	0.2	0.3	1.64044	0.20130	0.31577	0.05970	0.00087	0.00258	1.62416	0.20055	0.30922	0.03668	0.00036	0.00127
1.6	0.2	1.7	1.64080	0.19998	1.79482	0.06436	0.00002	0.08717	1.63232	0.20010	1.74988	0.03550	0.00001	0.04742
1.6	0.2	5.0	1.62814	0.20005	5.22257	0.05475	0.00000	0.72139	1.61006	0.20004	5.14097	0.03047	0.00000	0.41215
1.6	1.5	0.3	1.64172	1.51804	0.31409	0.05912	0.04400	0.00249	1.62863	1.50330	0.30960	0.03558	0.02118	0.00127
1.6	1.5	5.0	1.62901	1.50058	5.22638	0.05627	0.00014	0.71332	1.60478	1.49987	5.15953	0.02978	0.00007	0.38982
1.6	4.0	0.3	1.62918	4.06111	0.31276	0.05541	0.32855	0.00265	1.61808	3.99504	0.31052	0.03133	0.14399	0.00146
1.6	4.0	1.7	1.64360	4.00722	1.77624	0.05722	0.00951	0.08558	1.61299	3.99800	1.75432	0.03555	0.00426	0.04493
1.6	4.0	5.0	1.63797	4.00106	5.25027	0.06354	0.00108	0.71757	1.62001	4.00063	5.13799	0.03489	0.00061	0.41305
3.0	0.2	0.3	3.08367	0.22157	0.31638	0.23409	0.00405	0.00245	3.04353	0.21084	0.30855	0.14662	0.00169	0.00130
3.0	0.2	1.7	3.06657	0.20248	1.78674	0.23794	0.00009	0.08300	3.04115	0.20141	1.74705	0.14390	0.00004	0.04079
3.0	0.2	5.0	3.05929	0.20089	5.27383	0.22984	0.00001	0.70851	3.04622	0.20041	5.12599	0.13519	0.00001	0.36311
3.0	1.5	0.3	3.05429	1.64512	0.31480	0.23332	0.21787	0.00255	3.04415	1.57312	0.30713	0.13708	0.10655	0.00130
3.0	1.5	1.7	3.06052	1.51710	1.78322	0.26068	0.00480	0.08165	3.03280	1.50993	1.74986	0.13679	0.00268	0.04491
3.0	1.5	5.0	3.10828	1.50621	5.25946	0.23379	0.00060	0.74276	3.02710	1.50318	5.14911	0.13649	0.00031	0.39503
3.0	4.0	0.3	3.07690	4.39372	0.31468	0.22985	1.66922	0.00262	3.05295	4.20288	0.30719	0.13699	0.84081	0.00134
3.0	4.0	1.7	3.08967	4.05920	1.79784	0.24145	0.03831	0.08628	3.03756	4.02491	1.74494	0.13823	0.02043	0.04438
3.0	4.0	5.0	3.09394	4.01259	5.21241	0.24109	0.00396	0.64305	3.03010	4.00985	5.15383	0.13879	0.00216	0.37027

			n
			200						100
Parameters			Mean Values			MSE			Mean Values			MSE
$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$	$θ$	$α$	$β$
0.5	0.2	2.5	0.50246	0.19972	2.51801	0.00075	0.00005	0.02218	0.50004	0.19984	2.50568	0.00015	0.00001	0.00417
0.5	0.2	4.0	0.50037	0.19989	4.02296	0.00061	0.00002	0.05245	0.50034	0.19990	4.00866	0.00013	0.00000	0.01026
0.5	2.5	2.5	0.50087	2.49631	2.51720	0.00061	0.00707	0.02008	0.50069	2.50106	2.50053	0.00015	0.00145	0.00395
0.5	2.5	4.0	0.50291	2.49697	4.03103	0.00072	0.00304	0.05666	0.49990	2.50012	4.00303	0.00013	0.00059	0.01041
0.5	0.2	0.3	0.50202	0.20826	0.30125	0.00071	0.00416	0.00030	0.49999	0.20061	0.30055	0.00012	0.00069	0.00006
1.6	0.2	0.3	1.60415	0.19958	0.30308	0.00786	0.00008	0.00030	1.60189	0.19977	0.30090	0.00173	0.00001	0.00006
1.6	0.2	1.7	1.60751	0.20002	1.71108	0.00823	0.00000	0.01023	1.60040	0.20003	1.70097	0.00163	0.00000	0.00187
1.6	0.2	5.0	1.60839	0.19998	5.04457	0.00786	0.00000	0.07630	1.59866	0.20000	5.00736	0.00171	0.00000	0.01641
1.6	1.5	0.3	1.60704	1.49607	0.30331	0.00866	0.00439	0.00031	1.60120	1.50025	0.30041	0.00177	0.00080	0.00005
1.6	1.5	5.0	1.60249	1.49988	5.04088	0.00808	0.00002	0.07825	1.60078	1.50001	5.00577	0.00165	0.00000	0.01627
1.6	4.0	0.3	1.60858	4.00176	0.30213	0.00814	0.02899	0.00028	1.60095	3.99835	0.30055	0.00156	0.00570	0.00005
1.6	4.0	1.7	1.60470	4.00205	1.70443	0.00760	0.00098	0.00932	1.60187	3.99998	1.70279	0.00161	0.00018	0.00183
1.6	4.0	5.0	1.60656	3.99945	5.04988	0.00742	0.00011	0.08033	1.59865	3.99980	5.01257	0.00170	0.00002	0.01595
3.0	0.2	0.3	3.01580	0.20290	0.30230	0.03511	0.00034	0.00028	3.00156	0.20041	0.30031	0.00620	0.00006	0.00005
3.0	0.2	1.7	3.01093	0.20043	1.71386	0.03081	0.00001	0.00867	3.00384	0.20011	1.70353	0.00659	0.00000	0.00172
3.0	0.2	5.0	3.00480	0.20014	5.04237	0.02864	0.00000	0.08716	3.00362	0.20001	5.00335	0.00615	0.00000	0.01527
3.0	1.5	0.3	3.00938	1.51712	0.30173	0.03318	0.01950	0.00030	3.00637	1.50627	0.30073	0.00606	0.00380	0.00006
3.0	1.5	1.7	3.01056	1.50259	1.71168	0.02985	0.00057	0.00916	3.00947	1.50054	1.70237	0.00652	0.00010	0.00168
3.0	1.5	5.0	3.00359	1.50077	5.03344	0.03071	0.00006	0.07981	3.00540	1.50007	5.00299	0.00622	0.00001	0.01530
3.0	4.0	0.3	3.01337	4.07077	0.30298	0.03184	0.12498	0.00026	3.00193	4.01575	0.30066	0.00617	0.02279	0.00005
3.0	4.0	1.7	3.00841	4.00542	1.70934	0.03271	0.00448	0.01025	3.00348	4.00184	1.70300	0.00628	0.00081	0.00186
3.0	4.0	5.0	3.01074	4.00164	5.02749	0.03215	0.00046	0.07799	3.00249	3.99993	4.99994	0.00587	0.00009	0.01526

Distribution	Estimates				$W^{*}$	$A^{*}$
LiW( $θ, α, β$ )	2.5365	0.0399	0.8705		0.12115	0.70761
	(4.2430)	(0.0880)	(0.1100)
WLi( $θ, α, β$ )	0.0468	0.8027	0.0695		0.12278	0.71588
	(0.0400)	(0.1550)	(0.0270)
W( $α, β$ )	0.0860	0.9012			0.12568	0.72891
	(0.0120)	(0.0860)
WFr( $α, β, a, b$ )	0.1294	0.0681	2.2089	9.2457	0.12887	0.74176
	(0.8280)	(0.0110)	(10.115)	(4.7670)
TCWG( $α, β, γ, λ$ )	1.2612	0.6710	0.1458	-0.9102	0.16233	0.91437
	(1.0410)	(0.0470)	(0.0800)	(0.3450)
KwW( $α, β, a, b$ )	8.5878	0.1118	19.1940	56.9153	0.16421	0.92299
	(31.558)	(0.016)	(10.1530)	(79.9300)
BW( $α, β, a, b$ )	1.8533	0.2772	9.3774	1.7812	0.2346	1.30499
	(1.7000)	(0.0650)	(2.2240)	(2.1480)
GW( $α, β, γ$ )	2.5359	0.3397	1.4145		0.18747	1.0470
	(2.9470)	(0.1490)	(0.7730)

Distribution	Estimates				$W^{*}$	$A^{*}$
LiW( $θ, α, β$ )	0.3036	0.9209	1.8846		0.05022	0.49507
	(0.3138)	(0.6552)	(0.2465)
WLi( $θ, α, β$ )	0.2311	3.0359	0.3104		0.07666	0.54281
	(0.0819)	(0.4246)	(0.0214)
TCWG( $α, β, γ, λ$ )	0.0188	0.9598	1.4081	0.6649	0.07758	0.57770
	(0.0610)	(0.7160)	(2.7490)	(0.294)
W( $α, β$ )	0.3493	2.3744			0.08296	0.74365
	(0.0170)	(0.2100)
KwW( $α, β, a, b$ )	14.4331	0.2041	34.6599	81.8459	0.18523	1.50591
	(27.095)	(0.0420)	(17.5270)	(52.0140)
WFr( $α, β, a, b$ )	630.9384	0.3024	416.0971	1.1664	0.25372	1.95739
	(697.942)	(0.032)	(232.359)	(0.357)
BW( $α, β, a, b$ )	1.3600	0.2981	34.1802	11.4956	0.46518	3.21973
	(1.0020)	(0.0600)	(14.8380)	(6.7300)
GW( $α, β, γ$ )	2.3769	0.8481	3.5344		0.25533	1.94887
	(0.3780)	(0.0005)	(0.6650)

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[1] Correspondence to: Ahmed Zakarya Afify E-mail: ahmed.Afify@fcom.bu.edu.eg