The Nadarajah-Haghighi Lindley distribution

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

doi:10.1590/0001-3765201920170856

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Mathematical Sciences • An. Acad. Bras. Ciênc. 91 (01) • 2019 • https://doi.org/10.1590/0001-3765201920170856 copy

The Nadarajah-Haghighi Lindley distribution

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract: We define a new lifetime model based on compounding the Lindley and Nadarajah-Haghighi distributions. The proposed distribution is very competitive to other lifetime models. Some of its mathematical properties are investigated including generating function, mean residual life, moments, Bonferroni and Lorenz curves and mean deviations. We discuss the estimation of the model parameters by maximum likelihood. We provide a simulation study and two applications to real data for illustrative purposes. We prove empirically that the new distribution yields good fits to both data sets, and it can be a useful alternative for other classical lifetime models.

Key words
Compounding approach; Exponential distribution; Lifetime data; Lindley distribution; Nadarajah-Haghighi distribution

m(x)	α	λ	γ	Numerical integration	Series expansion
m(x)	α	λ	γ	Numerical integration	N=5	N=10
m(1)	1.2	1.5	0.5	0.3970	0.3962	0.3970
	1.5	2.0	1.5	0.1519	0.1081	0.1519
	2.5	1.5	0.5	0.0629	0.0629	0.0629

$μ'_{n}$	α	λ	γ	Numerical integration	Series expansion
$μ'_{n}$	α	λ	γ	Numerical integration	$N = 5$	$N = 10$	$N = 15$
$μ'_{1}$	1.2	1.5	0.5	0.45973	0.46241	0.45973	0.45973
	1.5	2.0	1.5	0.22280	0.23134	0.22278	0.22280
	2.5	1.5	0.5	0.18676	0.18677	0.18676	0.18676
$μ'_{2}$	1.2	1.5	0.5	0.39537	0.36964	0.39540	0.39537
	1.5	2.0	1.5	0.09018	0.06314	0.09031	0.09018
	2.5	1.5	0.5	0.05757	0.05756	0.05757	0.05757
$μ'_{3}$	1.2	1.5	0.5	0.48550	0.67914	0.48488	0.48550
	1.5	2.0	1.5	0.05092	0.11772	0.05013	0.05092
	2.5	1.5	0.5	0.02317	0.02327	0.02317	0.02317
$μ'_{4}$	1.2	1.5	0.5	0.76476	−0.28626	0.77456	0.76475
	1.5	2.0	1.5	0.03616	−0.08171	0.04024	0.03615
	2.5	1.5	0.5	0.01116	0.01057	0.01116	0.01116

m_s(x)	α	λ	γ	Numerical integration	Series expansion
m_s(x)	α	λ	γ	Numerical integration	$N = 2$	$N = 5$
m₁(0.5)	1.2	1.5	0.5	0.1380	0.1381	0.1380
	1.5	2.0	1.5	0.1559	0.1604	0.1559
	2.5	1.5	0.5	0.1609	0.1610	0.1609
m₁(1)	1.2	1.5	0.5	0.3072	0.3095	0.3072
	1.5	2.0	1.5	0.2166	0.2489	0.2161
	2.5	1.5	0.5	0.1866	0.1869	0.1866
m₂(0.5)	1.2	1.5	0.5	0.0421	0.0422	0.0421
	1.5	2.0	1.5	0.0421	0.0439	0.0421
	2.5	1.5	0.5	0.0417	0.0417	0.0417
m₂(1)	1.2	1.5	0.5	0.1663	0.1681	0.1663
	1.5	2.0	1.5	0.0829	0.1062	0.0825
	2.5	1.5	0.5	0.0574	0.0576	0.0574

Scenario	α	λ	γ	n	Mean			RMSE
Scenario	α	λ	γ	n	$\hat{α}$	$\hat{λ}$	$\hat{γ}$	$\hat{α}$	$\hat{λ}$	$\hat{γ}$
1	0.1	0.2	0.3	50	0.4136	0.1763	0.2710	0.7718	0.0435	0.0735
				100	0.2746	0.1842	0.2865	0.4211	0.0284	0.0520
				300	0.1684	0.1891	0.3006	0.1640	0.0127	0.0228
				500	0.1443	0.1896	0.3048	0.1048	0.0114	0.0188
2	0.5	1.8	1.1	50	0.5808	1.7829	1.1100	0.4232	0.0955	0.0360
				100	0.5474	1.7854	1.1107	0.1645	0.0757	0.0316
				300	0.5266	1.7886	1.1102	0.0418	0.0409	0.0196
				500	0.5212	1.7899	1.1098	0.0311	0.0101	0.0098
3	0.5	2.5	0.1	50	0.5361	2.4629	0.1033	0.0620	0.1472	0.0292
				100	0.5248	2.4749	0.1065	0.0387	0.0954	0.0171
				300	0.5159	2.4844	0.1086	0.0206	0.0411	0.0117
				500	0.5132	2.4863	0.1093	0.0158	0.0323	0.0110
4	0.7	2.8	0.1	50	0.8187	2.5700	0.1540	0.3410	0.5953	0.2359
				100	0.7586	2.6585	0.1259	0.1346	0.4069	0.1371
				300	0.7246	2.7570	0.1121	0.0427	0.1522	0.0578
				500	0.7176	2.7781	0.1101	0.0245	0.0646	0.0228
5	0.7	4	0.5	50	0.8298	3.7395	0.4800	0.5494	0.7593	0.1356
				100	0.7663	3.8218	0.4940	0.1629	0.5583	0.0774
				300	0.7340	3.8921	0.5151	0.0697	0.3781	0.0569
				500	0.7229	3.9379	0.5139	0.0438	0.2558	0.0476
6	0.8	0.4	0.1	50	0.9127	0.3675	0.1323	0.5489	0.0798	0.0737
				100	0.8593	0.3785	0.1209	0.33249	0.04984	0.04455
				300	0.8256	0.3859	0.1143	0.03713	0.02524	0.02618
				500	0.8200	0.3882	0.1115	0.0268	0.0191	0.0204
7	0.9	0.5	0.1	50	1.0241	0.4625	0.1387	0.4564	0.0931	0.0938
				100	0.9832	0.4719	0.1231	0.3026	0.0707	0.0600
				300	0.9294	0.4854	0.1154	0.0416	0.0251	0.0322
				500	0.9238	0.4868	0.1129	0.0401	0.0216	0.0268
8	1.3	0.4	0.1	50	1.5905	0.3703	0.1205	0.9387	0.0926	0.0671
				100	1.4492	0.3801	0.1244	0.7274	0.0602	0.0652
				300	1.3576	0.3860	0.1206	0.2268	0.0326	0.0475
				500	1.3351	0.3895	0.1154	0.0685	0.0129	0.0225

Statistics	Real data sets
Statistics	Reinfection times	Exceedances of flood peaks
Mean	369.5268	12.2042
Median	247.0000	9.5000
Mode	5.000	5.0000
Variance	136940.7000	151.2215
Skewness	1.15262	1.4725
Kurtosis	0.51036	2.8895
Maximum	1529.0000	64.0000
Minimum	1.0000	0.1000
$n$	877	72

Distributions	Estimates
NHL( $α, λ, γ$ )	0.1247	0.0718	0.0031
	(0.0121)	(0.0130)	(0.0001)
ENH( $α, λ, β$ )	0.8445	0.0030	0.7908
	(0.0347)	(0.0001)	(0.0324)
EW( $α, λ, β$ )	0.9819	0.0026	0.7447
	(0.0100)	(0.0001)	(0.0293)
Weibull( $α, λ$ )	0.8464	0.0073
	(0.0206)	(0.0009)
NH( $α, λ$ )	0.7950	0.0039
	(0.0363)	(0.0003)
Lindley( $γ$ )	0.0053
	(0.0001)
Exp( $λ$ )	0.0026
	(7.568e-05)

	Statistics
Distributions	AIC	CAIC	$W^{*}$	$A^{*}$	KS
NHL	12035.71	12035.74	0.0904	0.9826	0.0332
ENH	12081.29	12081.32	0.4790	3.6459	0.0477
EW	12069.46	12069.49	0.3422	2.7487	0.0576
Weibull	12084.77	12084.78	0.5082	3.8380	0.0552
NH	12121.64	12121.65	0.5782	4.2894	0.0677
Lindley	12828.07	12828.08	0.3532	2.8802	0.2076
Exp	12126.04	12126.04	0.3572	2.8505	0.0844

Distributions	Estimates
NHL( $α, λ, γ$ )	0.1693	0.9332	0.0898
	(0.0361)	(0.0190)	(0.0064)
ENH( $α, λ, β$ )	1.6963	0.0313	0.7302
	(1.1459)	(0.0307)	(0.1353)
EW( $α, λ, β$ )	1.3392	0.0190	0.5445
	(0.4069)	(0.0303)	(0.2400)
Weibull( $α, λ$ )	0.9019	0.1095
	(0.0858)	(0.0302)
NH( $α, λ$ )	0.8464	0.1079
	(0.2493)	(0.0558)
Lindley( $γ$ )	0.1530
	(0.0128)
Exp( $λ$ )	0.0819
	(0.0096)

	Statistics
Distributions	AIC	CAIC	$W^{*}$	$A^{*}$	KS
NHL	505.93	506.28	0.0897	0.5286	0.0979
ENH	507.84	508.20	0.1026	0.6275	0.1066
EW	508.05	508.40	0.1071	0.6504	0.1072
Weibull	506.99	507.17	0.1379	0.7851	0.1052
NH	507.97	508.15	0.1437	0.8149	0.1258
Lindley	530.42	530.48	0.1391	0.8526	0.2409
Exp	506.25	506.31	0.1306	0.7523	0.1422

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[1] Correspondence to: Fernando Arturo Peña-Ramírez E-mail: fernando.p.ramirez@ufsm.br ORCid: https://orcid.org/0000-0001-9605-1766