A weighted negative binomial Lindley distribution with applications to dispersed data

BAKOUCH, HASSAN S

doi:10.1590/0001-3765201820170733

HASSAN S BAKOUCH

Department of Mathematics, Faculty of Science, Tanta University, Tanta, 315277, Egypt

Correspondence to: Carlos Eduardo Ambrósio E-mail: ceambrosio@usp.br

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Figure 1
Probability Graphs for the Indicated Values of p and b of WNBL.

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Figure 2
Hazard Function Graphs for the Indicated Values of p and b of WNBL.

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TABLE I
Average Bias and MSE values of the MMEs from simulation of the WNBL distribution.

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TABLE II
Average Bias and MSE values of the MLEs from simulation of the WNBL distribution.

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Table III
Counts of numbers of European red mites on apple leaves; data from Garman, 1951.

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Table IV
Summary Statistics of data-I.

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TABLE V
Tests Summary for data set I.

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Table VI
Accidents to 647 women working on H. E. shells during 5 weeks.

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Table VII
Summary Statistics of data-II.

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TABLE VIII
Tests Summary for data set II.

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Table IX
Number of outbreaks of strikes in UK coal mining industries.

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Table X
Summary Statistics of Data-III.

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TABLE XI
Tests Summary for data set III.

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TABLE XII
Frequencies of the observed number of days that experience x TH’s at Cape Kennedy,Fla.,for 11-year period of Record January 1957 through December 1967.

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TABLE XIII
Expected Frequencies,Log Likelihood, x²,AIC and p-values of the observed number of days that experience x TH’s at Cape Kennedy,Fla.,from WNBDL.

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TABLE XIV
Expected Frequencies,Log Likelihood, x²,AIC and p-values of the Observed number of days that experience x TH’s at Cape Kennedy,Fla.,from NB.

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TABLE XV
Expected Frequencies,Log Likelihood, x², AIC and p-values of the observed number of days that experience x TH’s at Cape Kennedy,Fla.,from GP.

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Table XVI
Comparison via Vuong Test.

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Table XVII
Comparison via Vuong Test.

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Parameter	Sample Size	Bias( $\hat{β}$ )	Bias( $\hat{p}$ )	MSE( $\hat{β}$ )	MSE( $\hat{p}$ )
β = 0:0542, p = 0:7487	n=20	26.6700	-0.7485	742.1	0.5603
	n=50	14.5694	-0.6458	411.023	0.4276
	n=100	1.1141	-0.6022	1.4983	0.4265
	n=200	0.9135	-0.5743	0.9130	0.3924
	n=500	0.8783	-0.5517	0.8940	0.3822
β = 30:4389, p = 0:0157	n=20	139.4975	-0.7418	37071.8	0.5502
	n=50	4.7926	-0.7071	23.0087	0.5001
	n=100	4.4596	-0.6744	19.4900	0.4995
	n=200	3.8780	-0.6688	18.8034	0.4872
	n=500	2.8913	-0.6217	15.8204	0.4644
β= 5:2856, p = 0:6398	n=20	4.0410	-0.0942	12.6543	0.0125
	n=50	3.6904	-0.0907	12.1628	0.0063
	n=100	3.6522	-0.0890	9.8270	0.0058
	n=200	3.3172	-0.0836	8.5084	0.0056
	n=500	2.6516	-0.0798	8.2922	0.0047

Parameter	Sample Size	Bias( $\hat{β}$ )	Bias( $\hat{p}$ )	MSE( $\hat{β}$ )	MSE( $\hat{p}$ )
β = 0:0542, p = 0:7487	n=20	32.0968	-0.7486	1030.22	0.5604
	n=50	14.498	-0.6101	410.9645	0.3914
	n=100	0.9708	-0.5826	1.1641	0.3570
	n=200	0.7625	-0.5720	0.6627	0.3545
	n=500	0.6575	-0.5606	0.6321	0.3532
β = 30:4389, p = 0:0157	n=20	138.775	-0.7417	36685.5	0.5502
	n=50	4.1503	-0.6758	18.827	0.4968
	n=100	3.6311	-0.6408	17.574	0.4888
	n=200	3.3901	-0.6212	16.8538	0.4837
	n=500	2.3361	-0.5537	14.3888	0.4327
β = 5:2856, p = 0:6398	n=20	3.3960	-0.0869	11.5333	0.0075
	n=50	3.0078	-0.0866	11.3053	0.0055
	n=100	2.5571	-0.0866	10.3211	0.0052
	n=200	2.4807	-0.0752	7.7243	0.0045
	n=500	1.6978	-0.0728	5.1232	0.0042

S_{k} = \frac{(1 - p)^{β + 1} p^{k} (β)_{k}}{(1 + p (β^{2} - 1)) x k!} \times {(1 + β k)_{2} F_{1} (β + k, 1; k + 1; p) + \frac{β p (β + k)_{2} F_{1} (β + k + 1, 2; k + 2; p)}{(k + 1)}}

Count	0	1	2	3	4	5	6	7	Total
Observed frequency	70	38	17	10	9	3	2	1	150

Mean	Variance	ID	MDμ	Skewness	Kurtosis
1.1466	2.2736	1.9828	1.1445	1.5291	4.9528

Distribution	$\hat{p}$	$\hat{β}$	KS	- $l$	df	$χ^{2}$	p-value	AIC	BIC	AICc
WNBL( $\hat{p}$ , $\hat{β}$ )	0.2087(0.0154)	0.4770(0.0436)	0.5951	222.3063	3	2.3842	0.497	448.6126	454.6339	445.6926
NB( $\hat{p}$ , $\hat{β}$ )	0.5281(0.0275)	1.0245( 0.1051)	0.5979	222.4371	3	2.5147	0.473	448.8742	454.8955	445.9542
GP( $\hat{p}$ , $\hat{β}$ )	0.3209(0.0551)	0.7786(0.0796)	0.6019	222.7452	3	2.9475	0.400	449.4904	455.5117	446.5704
Pois.( $\hat{β}$ )		1.1466(0.0874)	0.6408	242.8099	2	26.6513	0.000	487.6198	490.6304	485.6465

Count	0	1	2	3	4	5 or more	Total
Observed frequency	447	132	42	21	3	2	647

Mean	Variance	ID	MDμ	Skewness	Kurtosis
0.4652	0.6919	1.4872	0.6428	2.1212	7.9209

Distribution	$\hat{p}$	$\hat{β}$	KS	- $l$	df	$χ^{2}$	p-value	AIC	BIC	AICc
WNBL( $\hat{p}$ , $\hat{β}$ )	0.3006(0.0140)	0.6160(0.0314)	0.7295	592.0683	2	2.9852	0.225	1188.137	1197.081	1185.155
NB( $\hat{p}$ , $\hat{β}$ )	0.3497(0.0137)	0.8651(0.0587)	0.7309	592.2670	2	3.2742	0.195	1188.534	1197.479	1185.553
GP( $\hat{p}$ , $\hat{β}$ )	0.1963(0.0332)	0.3739(0.0253)	0.7328	592.6005	2	3.7961	0.150	1189.201	1198.146	1186.22
Pois.( $\hat{β}$ )		0.4652(0.0268)	0.7534	617.1843	2	65.0106	0.000	1236.369	1240.841	1234.375

Count	0	1	2	3	4	Total
Observed frequency	46	76	24	9	1	156

Mean	Variance	ID	MDμ	Skewness	Kurtosis
0.9935	0.7419	0.7467	0.5859	0.8023	3.5038

Distribution	$\hat{p}$	$\hat{β}$	KS	- $l$	df	$χ^{2}$	p-value	AIC	BIC	AICc
WNBL( $\hat{p}$ , $\hat{β}$ )	0.0833(0.0077)	3.9294(0.2916)	0.5729	187.3976	1	1.1206	0.290	378.7952	384.8949	375.8721
NB( $\hat{p}$ , $\hat{β}$ )	0.0017(5.6903 $\times 10^{- 06}$ )	572.1488(45.9855)	0.5395	191.9723	1	9.4912	0.002	387.9446	394.0443	385.0215
GP( $\hat{p}$ , $\hat{β}$ )	-0.1450(0.0403)	1.1376(0.0779)	0.5487	188.8228	1	4.5304	0.033	381.6456	387.7453	378.7225
Pois.( $\hat{β}$ )		0.9936(0.0798)	0.5381	191.9362	2	9.4145	0.009	385.8724	388.9223	383.898

x	January	February	March	April	May	June	July	August	September	October	Nov.	Dec.	Spring	Summer	Fall
0	335	295	308	299	266	187	177	185	228	311	321	334	873	549	860
1	4	9	20	18	43	77	80	89	54	17	6	3	81	246	77
2	2	4	9	10	25	40	47	30	33	9	3	2	44	117	45
3		2	3	3	3	17	26	24	12	4		2	9	67	16
4			1		3	6	9	10	3				4	25	3
5					0	2	2	3					0	7
6					1	1							1	1
Total	341	310	341	330	341	330	341	341	330	341	330	341	1012	1012	1001

x	Jan.	Feb.	March	April	May	June	July	Aug.	Sep.	Oct.	Nov.	Dec.	Spring	Summer	Fall
0	334.99	294.92	307.72	298.62	265.09	185.42	173.55	184.55	225.47	310.66	320.97	333.98	871.38	543.54	857.62
1	4.63	10.34	22.56	21.67	49.00	83.09	91.84	87.20	65.33	20.41	6.95	4.197	93.70	262.13	93.87
2	0.98	2.92	6.64	6.16	16.57	36.22	43.18	39.67	24.21	6.06	1.48	1.43	28.96	119.04	29.91
3		1.06	2.42	2.16	6.25	15.14	18.98	17.26	9.25	2.25		0.65	10.69	51.37	11.41
4			0.96		2.45	6.14	8.00	7.28	3.55				4.24	21.43	4.67
5					0.98	2.44	3.28	3.01					1.74	8.73
6					0.39	0.96							0.73	3.49
Total	341	310	341	330	341	330	341	341	330	341	330	341	1012	1012	1001
LL( $- l$ )	34.86	75.33	142.11	133.79	259.57	394.33	439.78	423.75	317.00	133.19	48.29	43.62	549.79	1258.98	560.87
$\hat{p}$	0.399	0.499	0.461	0.442	0.401	0.354	0.358	0.367	0.361	0.477	0.386	0.659	0.439	0.360	0.455
$\hat{β}$	0.034	0.066	0.141	0.144	0.731	0.815	0.740	0.526	0.123	0.053	0.019	0.019	0.204	0.760	0.200
Var $(\hat{p})$	0.012	0.0051	0.0021	0.002	0.001	0.0003	0.0003	0.0003	0.0005	0.0023	0.0073	0.011	0.0005	0.0001	0.0005
Var $(\hat{β})$	0.0002	0.0002	0.0004	0.001	0.001	0.0019	0.0021	0.0018	0.0014	0.0004	0.0003	0.00003	0.0003	0.0006	0.0002
Cov $(\hat{β}, \hat{p})$	0.0002	0.0012	0.001	0.001	0.001	0.0008	0.0008	0.0008	0.0009	0.0011	0.0015	0.0009	0.0003	0.0003	0.0003
Sk. $((\hat{β}, \hat{p})$	161.49	85.3649	51.053	50.412	36.69	44.767	49.662	46.471	37.537	55.072	108.482	241.63	42.2	46.884	42.551
Kur. $((\hat{β}, \hat{p})$	217.87	110.767	59.456	58.598	34.25	32.069	33.828	32.770	30.661	65.549	142.418	343.444	45.438	32.851	45.680
ID $((\hat{β}, \hat{p})$	1.69	2.109	2.0204	1.946	1.860	1.640	1.6217	1.6728	1.7132	2.079	1.678	3.030	1.973	1.647	2.037
AIC	73.72	154.66	288.22	271.58	523.14	792.66	883.56	851.5	638	270.38	100.58	91.24	1103.58	2521.96	1125.74
$χ^{2}$	0.027	0.032	1.173	3.250	5.504	2.392	4.538	5.745	5.559	3.306	0.038	0.083	9.80	7.90	11.18
df	1	1	2	2	3	4	4	4	3	3	2	1	4	5	3
p-value	0.869	0.858	0.556	0.197	0.138	0.664	0.338	0.219	0.135	0.347	0.981	0.773	0.044	0.162	0.011

x	Jan.	Feb.	March	April	May	June	July	Aug.	Sep.	Oct.	Nov.	Dec.	Spring	Summer	Fall
0	334.99	294.94	307.79	298.71	265.09	184.64	171.83	183.86	225.14	310.75	320.97	333.98	871.68	540.51	858.12
1	4.62	10.32	22.55	21.62	49.39	84.55	94.06	88.66	29.33	20.37	6.94	4.195	93.83	267.18	93.81
2	0.98	2.91	6.58	6.09	16.22	35.87	43.32	39.28	23.74	6.01	1.48	1.43	28.52	118.32	29.42
3		1.06	2.39	2.15	6.09	14.82	18.69	16.90	9.00	2.23		0.65	10.54	50.37	11.23
4			0.96		2.44	6.04	7.79	7.16	3.51				4.24	21.01	4.67
5					0.43	2.44	3.18	3.01					1.79	8.66
6					0.39	0.98							0.78	3.54
Total	341	310	341	330	341	330	341	341	330	341	330	341	1012	1012	1001
LL( $- l$ )	34.86	75.34	142.17	133.86	259.78	394.59	440.18	423.95	317.57	133.26	48.29	43.62	550.14	1259.91	561.41
$\hat{p}$	0.412	0.528	0.510	0.492	0.471	0.391	0.374	0.404	0.422	0.524	0.405	0.659	0.500	0.391	0.518
$\hat{β}$	0.034	0.066	0.144	0.147	0.396	1.172	1.465	1.192	0.698	0.125	0.053	0.019	0.215	1.263	0.211
AIC	73.72	154.68	288.34	271.72	523.56	793.18	884.36	851.9	639.14	270.52	100.58	91.24	1104.28	2523.82	1126.82
$χ^{2}$	0.028	0.035	1.232	3.347	6.008	2.888	5.428	5.967	6.429	3.307	0.039	0.084	10.38	9.51	11.87
df	1	1	2	2	3	4	4	4	3	3	2	1	4	5	3
p-value	0.867	0.852	0.540	0.188	0.111	0.577	0.246	0.202	0.093	0.347	0.981	0.772	0.034	0.090	0.008

x	Jan.	Feb.	March	April	May	June	July	Aug.	Sep.	Oct.	Nov.	Dec.	Spring	Summer	Fall
0	334.98	294.01	307.65	298.55	264.57	183.74	170.46	183.04	224.09	310.59	320.96	333.97	857.081	537.41	857.08
1	4.68	12.18	23.18	22.22	50.81	86.27	96.14	90.36	68.25	21.01	7.04	4.42	97.04	272.77	97.04
2	0.92	2.59	6.17	5.73	15.59	35.53	43.26	38.89	23.26	5.62	1.39	1.30	27.83	117.51	27.83
3		0.77	2.23	2.01	5.76	14.40	18.28	16.43	8.62	2.06		0.57	10.45	49.04	10.45
4			0.96		2.44	6.04	7.79	7.16	3.51				4.47	20.42	4.47
5					0.43	2.44	3.18	3.01					8.55	8.73
6					0.39	0.98							3.61	3.61
Total	341	310	341	330	341	330	341	341	330	341	330	341	1012	1012	1001
LL( $- l$ )	34.94	75.88	142.57	134.35	260.26	394.98	440.81	424.34	318.36	133.79	48.41	43.92	551.65	1261.37	563.64
$\hat{p}$	0.241	0.245	0.312	0.297	0.279	0.221	0.207	0.231	0.239	0.323	0.236	0.453	0.303	0.221	0.316
$\hat{β}$	0.018	0.053	0.103	0.100	0.254	0.586	0.694	0.622	0.387	0.093	0.028	0.021	0.150	0.633	0.1552
AIC	73.88	155.76	289.14	272.7	524.52	793.96	885.62	852.68	640.72	271.58	100.82	91.84	1107.3	2526.74	1131.28
$χ^{2}$	0.028	0.022	1.879	4.385	7.167	3.553	6.556	6.496	7.867	4.440	0.039	0.081	13.32	11.91	15.87
df	1	1	2	2	3	4	4	4	3	3	2	1	4	5	3
p-value	0.867	0.882	0.391	0.112	0.067	0.470	0.161	0.165	0.049	0.218	0.981	0.776	0.010	0.036	0.001

Competing Models	Statistics	January	February	March	April	May	June
WNBL-NB	𝐙 p-value	83.5625	31.42032	36.5475	108.1294	7.2535	69.2805
WNBL-NB	𝐙 p-value	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
WNBL-GP	𝐙 p-value	3.4798	1.0256	12.113	15.0277	5.5650	40.252
WNBL-GP	𝐙 p-value	0.0005	0.3051	0.0001	0.0001	0.0001	0.0001

Competing Models	Statistics	July	August	September	October	November	December
WNBL-NB	𝐙 p-value	90.2142	125.0967	26.7266	183.6862	1032.339	1838.215
WNBL-NB	𝐙 p-value	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
WNBL-GP	𝐙 p-value	69.8226	54.8183	12.113	13.5131	6.1042	3.1211
WNBL-GP	𝐙 p-value	0.0005	0.0001	0.0001	0.0001	0.0001	0.0018

[1] Correspondence to: Carlos Eduardo Ambrósio E-mail: ceambrosio@usp.br

Brasil

Brasil

A weighted negative binomial Lindley distribution with applications to dispersed data

Abstract