Table 4. Comparisons of the ZIP, ZINB, and ZICMP models with different dependence structures.

Marginal	Copula	Order	$\mathrm{A}\mathrm{I}\mathrm{C}$a	BICb	RMSPEc
ZIP	Gaussian	1	435.55	945.76	1.59
	Frank	1	434.11	942.88	1.58
	Gumbel	1	438.32	951.32	1.60
	ref.Gumbel	1	433.87	942.41	1.57
	Plackett	1	435.21	945.09	1.58
	Gaussian	2	433.85	950.22	1.56
	PSLT/Frank	2	431.25	948.87	1.56
	LSLT/Frank	2	433.77	953.90	1.55
	PSLT/Gumbel	2	418.46	923.28	1.62
ZINB	Gaussian	1	425.86	938.08	1.59
	Frank	1	424.54	935.45	1.59
	Gumbel	1	428.20	942.77	1.60
	ref.Gumbel	1	424.15	934.67	1.58
	Plackett	1	425.5	937.38	1.59
	Gaussian	2	424.06	942.34	1.56
	PSLT/Frank	2	417.00	928.23	1.57
	LSLT/Frank	2	423.67	941.56	1.57
	PSLT/Gumbel	2	413.05	920.32	1.63
ZICMP	Gaussian	1	477.10	1040.56	1.91
	Frank	1	475.06	1036.49	1.88
	Gumbel	1	473.49	1033.3	1.79
	ref.Gumbel	1	473.73	1033.84	1.87
	Plackett	1	469.06	1024.48	1.77
	Gaussian	2	449.97	994.16	1.72
	PSLT/Frank	2	457.44	1009.11	1.80
	LSLT/Frank	2	450.91	996.04	1.77
	PSLT/Gumbel	2	455.12	1004.47	1.83
Poisson	Frank	1	495.90	1038.9	1.76
	ref.Gumbel	1	488.16	1023.43	1.75
NB	Frank	1	446.28	947.52	2.32
	ref.Gumbel	1	445.69	946.35	2.29

^a AIC $=l(\hat{\mathit{\vartheta }};\mathbf{y})-r$,

^b BIC $=-2l(\hat{\mathit{\vartheta }};\mathbf{y})+r\log n$.

^c $\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{P}\mathrm{E}=\left\{1/n-p\sum _{t=1+p}^n\right[y_t-\hat{\mathrm{E}}(Y_t|Y_{t-1}=y_{t-1},\dots ,Y_{t-p}=y_{t-p};\hat{\mathit{\vartheta }}){]}^2{\}}^{1/2}.$