Flexible Lévy-Based Models for Time Series of Count Data with Zero-Inflation, Overdispersion, and Heavy Tails

Table 1

Summary statistics for the MM estimator for different parameter values , using an exponential kernel (representing short-range dependence) at different sample lengths of the series for a Poisson-inverse Gaussian Lévy-based process.

N	True value
		1.9	5.5	0.03	2	9	0.04	9	0.8	6.7
30	Mean	1.8666	5.4666	0.0288	1.9666	8.9666	0.0388	8.9666	0.7666	6.6988
	Bias	0.0022	−0.0004	−0.0000	0.0055	−0.0011	−0.0153	−0.0011	−0.1744	0.2066
	MSE	0.0001	0.0000	0.0000	0.0009	0.0000	0.0070	0.0000	0.9129	1.2808
100	Mean	1.8900	5.4900	0.0299	1.9900	8.9900	0.0399	8.9900	0.7900	6.6999
	Bias	0.0009	−0.0001	−0.0000	0.0019	−0.0001	−0.0046	−0.0001	−0.0521	0.0619
	MSE	0.0000	0.0000	0.0000	0.0003	0.0000	0.0021	0.0000	0.2714	0.3843
400	Mean	1.8975	5.4975	0.0300	1.9975	8.9975	0.0399	8.9975	0.7975	6.6999
	Bias	0.0002	−0.0000	0.0000	0.0004	−0.0000	−0.0011	−0.0000	−0.0130	0.0154
	MSE	0.0000	0.0000	0.0000	0.0000	0.0000	0.0005	0.0000	0.0676	0.0960
N	True value	7	0.4	0.05	10	0.7	3	3	6	9
100	Mean	6.9900	0.3900	0.0499	9.9900	0.6900	2.9999	2.9900	5.9900	8.9999
	Bias	0.0519	−0.0561	−0.0045	0.0819	−0.0531	0.0249	0.0119	−0.0001	0.0849
	MSE	0.2693	0.3147	0.0020	0.6707	0.2819	0.0624	0.0141	0.0000	0.7224
200	Mean	6.9950	0.3950	0.0499	9.9950	0.6950	2.9999	2.9950	5.9950	8.9999
	Bias	0.0259	−0.0280	−0.0022	0.0409	−0.0265	0.0124	0.0059	−0.0000	0.0424
	MSE	0.1349	0.1570	0.0010	0.3357	0.1407	0.0312	0.0071	0.0000	0.3612
400	Mean	6.9975	0.3975	0.0499	9.9975	0.6975	2.9999	2.9975	5.9975	8.9999
	Bias	0.0129	−0.0140	−0.0011	0.0204	−0.0132	0.0062	0.0029	−0.0000	0.0212
	MSE	0.0675	0.0784	0.0005	0.1679	0.0702	0.0156	0.0035	0.0000	0.1806