Exponential dispersion models for overdispersed zero-inflated count data

Figures & data

Figure 1. Probability mass functions of the ABM class for $r=4,m=5,$ and p = 1, 3, and 5, respectively.

Figure 2. Probability mass functions of the ABM class for $r=4,m=5,$ and $p=6.5,10,$ and 15, respectively.

Figure 3. Probability mass functions of the LM class for $r=4,m=2,$ and $p=2.1,4,$ and 8, respectively.

Figure 4. Probability mass functions of the LM class for $r=4,m=2,$ and p = 12, 16, and 20, respectively.

Table 1. Characteristics of data sets.

	p0	$\overline{x}$	D	b1	b2
Data set 1	0.9298	0.08650	1.417	5.316	41.01
Data set 2	0.7150	0.4831	2.092	3.163	16.34
Data set 3	0.5909	1.391	4.394	2.230	7.807
Data set 4	0.3322	1.708	2.148	1.707	6.904

Table 2. Data set 1.

Value	0	1	2	3	4	5
Frequency	3719	232	38	7	3	1

Table 3. Performance of fitting data set 1.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–1183.52	0.543789	2	0.761935	1.760396	1.9866e-04
NLD	– 1183.97	2.312184	2	0.314714	2.235378	3.1047e-04
NGDP	–1183.44	0.383445	2	0.825536	1.303073	1.7734e-04
EDLID	– 1184.48	3.651826	2	0.161071	3.000701	4.3696e-04
BTD	–1186.20	9.251567	2	0.009796	4.056789	8.6835e-04
ZIP	–1187.78	14.771480	2	0.000620	5.313975	1.2621e-03
PTD($\gamma =2.684$)	–1183.39	0.437403	2	0.803561	1.053916	1.6459e-04
ABM(r = 10)	–1183.37	0.444316	2	0.800789	0.726210	1.5896e-04
LM(r = 4)	–1183.41	0.382719	2	0.825835	1.043140	1.6972e-04

Table 4. Data set 2.

Value	0	1	2	3	4	5	6	7	8
Frequency	296	74	26	8	4	4	1	0	1

Table 5. Performance of fitting data set 2.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–381.92	0.790146	3	0.851822	1.505635	6.0370e-03
NLD	– 382.13	2.292750	3	0.513911	1.732206	6.5419e-03
NGDP	–381.82	1.302363	3	0.728572	1.251828	5.7786e-03
EDLID	– 382.97	4.198396	3	0.240822	2.585325	8.5663e-03
BTD	–386.46	11.293501	3	0.010240	3.786089	1.4734e-02
ZIP	–391.63	27.329394	3	0.000005	5.714534	2.9491e-02
PTD($\gamma =2.920$)	–381.88	0.784300	3	0.853218	1.394523	5.9263e-03
ABM(r = 10)	–381.80	0.891831	3	0.827399	1.036148	5.7331e-03
LM(r = 3)	–381.95	0.753439	3	0.860568	1.297283	6.0981e-03

Table 6. Data set 3.

Value	0	1	2	3	4	5	6	7	8	9	10	11
Frequency	65	14	10	6	4	2	2	2	1	1	1	2

Table 7. Performance of fitting data set 3.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–170.87	4.754522	3	0.190680	2.762842	6.3329e-02
NLD	– 167.52	0.705282	3	0.871961	1.038579	3.2884e-02
NGDP	–168.87	2.530106	3	0.469875	2.243219	4.5103e-02
EDLID	– 166.95	0.151495	3	0.985011	0.665346	2.7689e-02
BTD	–170.91	4.485663	3	0.213573	2.108909	6.3666e-02
ZIP	–182.24	15.768584	3	0.001265	3.238417	1.6673e-01
PTD(γ = 2)	–167.54	0.632639	3	0.888921	1.034530	3.3090e-02
ABM(r = 1)	–167.54	0.632639	3	0.888921	1.034530	3.3090e-02
LM(r = 10)	–171.01	4.762802	3	0.190013	2.717090	6.4574e-02

Table 8. Data set 4.

Value	0	1	2	3	4	5	6	7	8	9	10	11
Frequency	99	65	57	35	20	10	4	0	3	4	0	1

Table 9. Performance of fitting data set 4.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–530.10	6.659803	4	0.154996	6.194263	3.6346e-02
NLD	– 528.40	2.408122	4	0.661160	3.527420	3.0641e-02
NGDP	–531.46	10.267733	4	0.036151	5.863542	4.0934e-02
EDLID	– 528.56	3.029695	4	0.552868	3.734667	3.1208e-02
BTD	–529.93	1.025681	4	0.905877	2.292173	2.9729e-02
ZIP	–542.67	12.091699	4	0.016682	5.288587	7.8550e-02
PTD(γ = 2)	–528.77	3.680797	4	0.450925	4.247745	3.1895e-02
ABM(r = 1)	–528.77	3.680797	4	0.450925	4.247745	3.1895e-02
LM(r = 10)	–529.65	5.682523	4	0.224145	5.674060	3.4849e-02

Table 10. Three sets of simulated data.

Value	0	1	2	3	4	5	6	7	8
Data set 5	8203	1417	290	62	19	5	0	3	1
Data set 6	7933	1420	406	132	63	23	16	4	3
Data set 7	4624	2387	1390	752	412	211	103	74	47

Table 11. Characteristics of simulated data sets.

	p0	$\overline{x}$	D	b1	b2
Data set 5	0.8203	0.2313	1.401	3.481	22.25
Data set 6	0.7933	0.3143	1.843	3.569	20.40
Data set 7	0.4624	1.164	2.009	1.702	6.126

Table 12. Performance of fitting data set 5.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–5931.76	1.035150	3	0.792748	4.648578	5.4929e-04
NLD	– 5936.02	5.965423	3	0.113305	9.971059	9.7550e-04
NGDP	–5934.05	3.220770	3	0.358823	9.859480	7.7833e-04
EDLID	– 5939.82	12.760586	3	0.005184	14.395989	1.3553e-03
BTD	–5947.82	29.654061	3	0.000002	19.082800	2.0303e-03
ZIP	–5966.77	95.916546	3	0.000000	29.550924	4.0501e-03
PTD($\gamma =2.812$)	–5931.90	0.909689	3	0.823089	3.483411	5.6369e-04
ABM(r = 10)	–5932.31	1.198908	3	0.753266	4.490144	6.0485e-04
LM(r = 2)	–5931.53	0.723555	3	0.867650	3.297800	5.2608e-04

Table 13. Performance of fitting data set 6.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–7106.28	6.023648	4	0.197389	19.786444	7.9821e-04
NLD	– 7107.42	15.922941	4	0.003124	20.825278	9.1194e-04
NGDP	–7102.73	2.192939	4	0.700322	4.990708	4.4282e-04
EDLID	– 7117.12	40.313621	4	0.000000	34.229900	1.8821e-03
BTD	–7149.64	147.108365	4	0.000000	52.296407	5.1338e-03
ZIP	–7207.74	510.837458	4	0.000000	77.450065	1.0944e-02
PTD($\gamma =2.513$)	–7102.73	1.732954	4	0.784723	5.734922	4.4270e-04
ABM(r = 6)	–7103.02	1.669665	4	0.796220	5.126902	4.7231e-04
LM(r = 10)	–7104.07	2.311162	4	0.678738	9.632771	5.7683e-04

Table 14. Performance of fitting data set 7.

Model	L	${\chi }^2$	df	p-value	RMSE	KL
PIG	–14995.75	99.083792	5	0.000000	149.291054	1.0650e-02
NLD	– 14934.82	11.271396	5	0.046257	52.119898	4.5567e-03
NGDP	–14936.26	10.581946	5	0.060329	40.518667	4.7004e-03
EDLID	– 14924.71	2.381891	5	0.794168	13.910594	3.5448e-03
BTD	–14946.95	74.021500	5	0.000000	78.646023	5.7693e-03
ZIP	–15205.17	968.884983	5	0.000000	221.474368	3.1591e-02
PTD(γ = 2)	–14935.23	11.539940	5	0.041665	51.234138	4.5976e-03
ABM(r = 1)	–14935.23	11.539940	5	0.041665	51.234138	4.5976e-03
LM(r = 10)	–14980.33	71.569157	5	0.000000	126.322372	9.1075e-03