Efficient accounting for estimation uncertainty in coherent forecasting of count processes

Figures & data

Figure 1. Poi-INAR(1) DGP with $\mu =5$, $\alpha =0.50$, and sample size T. Distribution of forecasts conditioned on value 5: conditional median (‘PF${}_{50}$’, true value: 5), 95%-quantile (‘PF${}_{95}$’, true value: 8), and limits l, u of 90%-PI (‘PI${}_{90}$’, true limits: 2, 8). Distribution under estimation uncertainty as gray dots, and boxplots of resampled distributions.

Figure 2. Poi-INAR(1) DGP with $\mu =5$, $\alpha =0.75$, and sample size T. Distribution of forecasts conditioned on value 5: conditional median (‘PF${}_{50}$’, true value: 5), 95%-quantile (‘PF${}_{95}$’, true value: 7), and limits l, u of 90%-PI (‘PI${}_{90}$’, true limits: 3, 7). Distribution under estimation uncertainty as gray dots, and boxplots of resampled distributions.

Figure 3. Strikes counts discussed in Section 4: time series plot (left) and sample ACF with time lag k (right).

Figure 4. Strikes counts in 2003 (solid line) together with different types of forecasts: conditional median (“PF${}_{50}$”), 95%-quantile (“PF${}_{95}$”), and 90%-PI (“PI${}_{90}$”), computed from ML-estimated parameter (upper panel) or based on resampled ML-estimates (lower panel).

Figure 5. Forecast PMFs $P(X_{T+1}=x|x_T)$ for strikes counts in Jan./Feb. 2003, computed from ML-estimated parameter (grey line) or based on resampled ML-estimates (box plots).

Table A1. Coherent forecasting of Poi-INAR(1) DGP: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’), computed from true parameter, ML-estimated parameter, or based on resampled ML-estimates.

						asymptotic resampling						parametric bootstrap
			agreement true vs. estimated			agreement estimated vs. resampling mode			true covered by resampling			agreement estimated vs. resampling mode			true covered by resampling
μ	α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
2.5	0.25	100	827	766	703	985	968	958	999	1000	1000	989	974	969	1000	999	1000
		250	880	861	854	995	986	991	1000	1000	1000	995	985	983	1000	1000	1000
		500	919	923	912	993	994	994	1000	1000	1000	993	997	993	1000	1000	1000
2.5	0.50	100	848	789	805	990	982	974	1000	1000	1000	988	989	977	999	1000	999
		250	938	863	910	995	992	994	1000	1000	1000	999	991	994	1000	1000	1000
		500	974	887	950	997	990	995	1000	999	1000	999	993	997	1000	1000	1000
2.5	0.75	100	939	853	703	996	981	957	1000	1000	999	995	985	975	1000	1000	999
		250	957	925	815	994	991	974	1000	1000	998	994	993	977	1000	1000	999
		500	985	959	871	998	998	991	1000	1000	1000	1000	1000	991	1000	1000	1000
5	0.25	100	759	689	649	984	973	904	1000	1000	1000	988	980	914	1000	1000	1000
		250	876	797	776	992	980	961	1000	1000	1000	992	989	961	1000	1000	1000
		500	920	878	835	992	998	977	1000	999	999	996	995	977	999	1000	1000
5	0.50	100	833	734	559	988	973	944	1000	998	997	993	974	940	1000	999	997
		250	872	814	644	992	984	958	1000	1000	997	996	991	956	1000	999	996
		500	941	885	706	996	995	974	1000	1000	999	1000	996	965	1000	999	999
5	0.75	100	869	769	639	988	974	959	1000	1000	998	993	984	963	1000	1000	1000
		250	918	875	771	997	979	979	1000	999	999	997	983	978	1000	999	1000
		500	955	882	832	995	990	986	1000	1000	999	996	991	987	1000	1000	1000
10	0.25	100	678	635	520	964	949	923	998	1000	996	977	963	933	998	999	998
		250	790	734	696	990	985	964	1000	1000	999	983	987	967	999	1000	999
		500	848	812	751	987	985	983	1000	999	1000	991	987	979	1000	998	999
10	0.50	100	728	646	465	987	956	928	999	999	993	988	965	928	999	998	995
		250	820	771	615	992	981	945	999	1000	996	994	986	941	999	1000	997
		500	887	859	671	991	992	955	1000	998	999	992	990	956	1000	999	997
10	0.75	100	823	719	546	988	971	929	1000	999	995	991	983	928	999	999	998
		250	890	816	671	998	988	954	1000	1000	998	997	988	961	1000	1000	1000
		500	916	876	805	994	991	977	1000	1000	998	995	990	978	1000	1000	999

Table A2. Coherent forecasting of Poi-INAR(1) DGP based on resampled ML-estimates: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’). Mean number of different forecasts obtained from resampling.

		$\mu =2.5$			$\mu =5$			$\mu =10$
α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
0.25	100	2.22	2.83	3.62	2.77	3.38	5.12	3.45	4.14	6.19
	250	1.76	2.16	2.66	2.10	2.48	3.42	2.59	2.98	3.92
	500	1.55	1.79	2.19	1.76	2.06	2.72	2.11	2.39	2.98
0.5	100	2.09	2.60	3.92	2.54	3.15	5.29	3.12	3.82	7.72
	250	1.69	2.01	2.60	1.96	2.30	3.44	2.31	2.77	4.69
	500	1.50	1.70	1.90	1.64	1.93	2.63	1.93	2.23	3.36
0.75	100	1.64	2.31	3.38	2.05	2.60	4.80	2.50	3.16	6.90
	250	1.36	1.82	2.58	1.65	2.03	3.09	1.92	2.37	4.13
	500	1.26	1.49	2.14	1.46	1.78	2.43	1.66	1.97	3.04

Table A3. Coherent forecasting of different count DGPs: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’), computed from true parameter, ML-estimated parameter, or based on resampled ML-estimates. Numbers of agreement of true and estimated forecasts.

DGP:			Poi-INAR(1)			Poi-INARCH(1)			NB-INAR(1) with ${\sigma }^2/\mu =1.5$			ZIP-INAR(1) with ${\sigma }^2/\mu =1.5$			Poi-INAR(2) with ${\alpha }_2=0.25$			BAR(1) with $\pi =0.25$
μ	α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
2.5	0.25	100	827	766	703	818	746	756	807	543	615	770	714	658	819	703	629	854	834	792
		250	880	861	854	906	806	842	872	718	745	879	847	772	870	823	762	908	889	906
		500	919	923	912	926	897	898	886	787	804	931	907	841	914	851	818	953	925	966
2.5	0.5	100	848	789	805	781	766	714	784	612	659	765	689	671	797	733	707	920	867	777
		250	938	863	910	860	863	813	864	721	806	831	833	766	884	825	811	981	956	862
		500	974	887	950	918	910	878	859	825	903	871	895	813	939	886	884	997	982	914
2.5	0.75	100	939	853	703	819	720	786	932	638	653	914	696	706	823	778	677	944	866	718
		250	957	925	815	862	846	873	943	764	807	936	791	812	910	860	826	961	904	799
		500	985	959	871	919	876	918	943	813	891	953	852	863	936	882	838	975	949	832
5	0.25	100	759	689	649	762	676	655	720	492	430	732	674	515	709	644	604	792	758	644
		250	876	797	776	845	784	795	848	670	614	807	789	699	823	772	759	879	854	803
		500	920	878	835	909	848	840	891	754	696	877	839	809	894	840	828	926	903	830
5	0.5	100	833	734	559	754	674	650	789	525	458	720	661	521	735	650	580	796	793	642
		250	872	814	644	873	781	694	903	693	631	838	798	647	842	779	721	834	883	741
		500	941	885	706	954	827	742	957	788	721	902	849	730	898	852	815	877	943	752
5	0.75	100	869	769	639	769	738	704	847	607	571	841	738	611	762	708	623	901	813	718
		250	918	875	771	863	858	787	901	737	716	893	812	747	853	830	759	921	883	830
		500	955	882	832	930	936	877	938	820	790	905	858	798	910	894	799	936	924	865
10	0.25	100	678	635	520	686	563	616	609	382	255	690	633	164	586	540	480	743	672	653
		250	790	734	696	790	727	715	763	575	446	814	785	303	762	712	690	808	822	755
		500	848	812	751	860	804	790	853	684	568	838	822	409	844	773	733	917	877	847
10	0.5	100	728	646	465	649	587	592	691	440	313	677	618	343	607	551	477	724	687	532
		250	820	771	615	808	714	688	809	609	473	808	753	535	771	692	677	815	809	660
		500	887	859	671	879	795	703	887	706	597	841	826	689	844	813	771	836	824	705
10	0.75	100	823	719	546	673	517	614	761	535	451	693	643	486	681	596	487	824	757	556
		250	890	816	671	804	610	786	857	676	620	832	790	675	814	777	625	885	852	649
		500	916	876	805	870	680	855	906	775	732	864	852	790	870	815	745	895	899	722

Table A4. Coherent forecasting of different count DGPs: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’), computed from true parameter, ML-estimated parameter, or based on resampled ML-estimates. Numbers of agreement of estimated forecasts and modes of resampled forecasts.

DGP:			Poi-INAR(1)			Poi-INARCH(1)			NB-INAR(1) with ${\sigma }^2/\mu =1.5$			ZIP-INAR(1) with ${\sigma }^2/\mu =1.5$			Poi-INAR(2) with ${\alpha }_2=0.25$			BAR(1) with $\pi =0.25$
μ	α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
2.5	0.25	100	985	968	958	989	979	973	983	959	947	982	975	965	979	953	921	989	986	974
		250	995	986	991	990	989	991	995	972	983	992	986	989	994	986	963	995	992	989
		500	993	994	994	997	995	992	994	989	986	993	994	995	994	985	978	994	992	997
2.5	0.5	100	990	982	974	983	988	969	985	944	946	990	977	958	980	942	919	986	982	979
		250	995	992	994	992	992	990	986	983	969	990	988	982	991	977	989	995	995	989
		500	997	990	995	993	993	990	987	981	988	996	992	990	998	993	984	1000	1000	997
2.5	0.75	100	996	981	957	976	951	943	994	943	925	994	962	962	978	948	917	996	978	975
		250	994	991	974	985	977	990	997	973	967	998	981	975	995	978	969	997	990	988
		500	998	998	991	994	986	987	996	986	994	995	986	978	992	989	973	1000	996	994
5	0.25	100	984	973	904	980	971	922	980	952	902	984	977	916	967	960	908	986	980	969
		250	992	980	961	991	988	986	982	982	956	989	988	966	991	976	971	991	990	987
		500	992	998	977	995	993	995	993	989	969	992	991	984	990	987	985	993	986	995
5	0.5	100	988	973	944	971	967	932	983	921	891	971	981	914	968	943	887	988	982	969
		250	992	984	958	986	992	977	990	976	956	984	989	968	994	975	939	992	987	974
		500	996	995	974	995	990	988	997	987	973	983	987	975	993	986	978	993	997	978
5	0.75	100	988	974	959	967	958	919	988	946	925	988	980	943	974	931	903	994	976	968
		250	997	979	979	980	984	970	988	976	963	988	993	978	987	980	966	999	989	987
		500	995	990	986	991	990	983	996	991	986	996	995	985	995	977	976	997	989	995
10	0.25	100	964	949	923	970	951	900	966	911	833	980	972	701	939	920	858	977	967	947
		250	990	985	964	983	986	967	980	956	899	989	991	806	982	963	939	988	986	975
		500	987	985	983	993	992	981	990	985	939	994	992	875	993	982	970	998	991	982
10	0.5	100	987	956	928	972	944	886	969	902	810	973	956	824	959	903	857	984	977	917
		250	992	981	945	986	981	971	979	952	908	990	991	910	979	965	935	985	984	962
		500	991	992	955	992	986	978	993	972	941	989	992	954	991	989	965	989	988	971
10	0.75	100	988	971	929	945	918	899	977	926	859	972	957	888	957	891	866	984	976	931
		250	998	988	954	982	982	965	988	972	932	983	983	946	976	970	937	997	989	965
		500	994	991	977	995	979	982	997	983	956	992	992	968	989	976	965	995	993	977

Table A5. Coherent forecasting of different count DGPs: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’), computed from true parameter, ML-estimated parameter, or based on resampled ML-estimates. Numbers of cases, where true forecasts covered by resampled forecasts.

DGP:			Poi-INAR(1)			Poi-INARCH(1)			NB-INAR(1) with ${\sigma }^2/\mu =1.5$			ZIP-INAR(1) with ${\sigma }^2/\mu =1.5$			Poi-INAR(2) with ${\alpha }_2=0.25$			BAR(1) with $\pi =0.25$
μ	α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
2.5	0.25	100	999	1000	1000	1000	1000	1000	1000	996	995	998	1000	999	1000	999	992	999	1000	1000
		250	1000	1000	1000	1000	999	1000	999	1000	1000	1000	1000	999	1000	1000	999	1000	1000	1000
		500	1000	1000	1000	1000	999	1000	1000	998	998	1000	1000	1000	1000	999	1000	1000	1000	1000
2.5	0.5	100	1000	1000	1000	999	1000	999	1000	996	999	998	999	998	1000	999	999	1000	1000	1000
		250	1000	1000	1000	1000	1000	1000	1000	997	1000	1000	999	1000	1000	998	1000	1000	1000	1000
		500	1000	999	1000	999	999	1000	1000	999	1000	1000	999	1000	1000	1000	1000	1000	1000	1000
2.5	0.75	100	1000	1000	999	1000	999	1000	1000	993	999	999	997	999	999	998	999	1000	999	999
		250	1000	1000	998	1000	1000	1000	1000	999	998	999	999	1000	1000	1000	999	1000	1000	1000
		500	1000	1000	1000	1000	999	1000	1000	1000	996	1000	1000	1000	999	1000	1000	1000	1000	1000
5	0.25	100	1000	1000	1000	998	999	1000	1000	993	990	999	999	994	1000	999	1000	1000	1000	1000
		250	1000	1000	1000	1000	998	999	999	999	1000	999	1000	1000	1000	999	1000	1000	1000	999
		500	1000	999	999	1000	1000	1000	1000	999	999	999	1000	1000	1000	1000	1000	1000	1000	1000
5	0.5	100	1000	998	997	999	998	999	999	996	994	999	995	993	1000	995	994	1000	999	997
		250	1000	1000	997	999	999	998	1000	1000	999	1000	1000	997	998	999	998	1000	1000	997
		500	1000	1000	999	1000	1000	1000	1000	1000	1000	1000	1000	998	1000	998	1000	999	1000	1000
5	0.75	100	1000	1000	998	999	1000	999	1000	996	998	996	996	996	1000	999	998	1000	1000	998
		250	1000	999	999	1000	1000	1000	1000	998	997	1000	1000	999	998	999	999	1000	1000	999
		500	1000	1000	999	1000	1000	999	1000	1000	999	1000	1000	996	1000	1000	997	1000	1000	999
10	0.25	100	998	1000	996	1000	1000	1000	999	991	987	998	996	951	1000	1000	997	999	999	998
		250	1000	1000	999	999	999	1000	1000	1000	996	1000	1000	991	998	1000	997	1000	1000	1000
		500	1000	999	1000	1000	1000	1000	1000	1000	1000	1000	1000	993	999	1000	1000	1000	1000	999
10	0.5	100	999	999	993	1000	999	998	1000	1000	992	991	991	982	1000	999	998	1000	999	997
		250	999	1000	996	999	1000	999	1000	998	999	998	1000	995	999	999	1000	1000	1000	999
		500	1000	998	999	1000	1000	998	1000	1000	997	1000	1000	1000	999	998	999	1000	1000	1000
10	0.75	100	1000	999	995	995	996	996	999	997	994	987	979	970	1000	1000	998	999	1000	995
		250	1000	1000	998	1000	998	1000	1000	999	999	997	997	998	1000	998	998	1000	1000	997
		500	1000	1000	998	1000	1000	1000	1000	1000	1000	1000	1000	1000	1000	999	998	1000	1000	1000

Table A6. Coherent forecasting of different count DGPs: conditional median (‘PF${}_{50}$’), 95%-quantile (‘PF${}_{95}$’), and 90%-PI (‘PI${}_{90}$’), computed from true parameter, ML-estimated parameter, or based on resampled ML-estimates. Mean lengths of support of resampled forecasts.

DGP:			Poi-INAR(1)			Poi-INARCH(1)			NB-INAR(1) with ${\sigma }^2/\mu =1.5$			ZIP-INAR(1) with ${\sigma }^2/\mu =1.5$			Poi-INAR(2) with ${\alpha }_2=0.25$			BAR(1) with $\pi =0.25$
μ	α	T	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$	PF${}_{50}$	PF${}_{95}$	PI${}_{90}$
2.5	0.25	100	2.22	2.83	3.62	2.27	2.98	3.04	2.53	4.18	4.51	2.72	3.28	3.94	2.50	3.20	4.34	2.15	2.32	3.04
		250	1.76	2.16	2.66	1.82	2.24	2.34	1.90	3.10	3.11	2.13	2.45	2.71	1.95	2.36	3.06	1.81	1.82	2.32
		500	1.55	1.79	2.19	1.56	1.88	1.95	1.67	2.46	2.43	1.77	2.02	2.07	1.67	1.98	2.45	1.54	1.61	1.89
2.5	0.5	100	2.09	2.60	3.92	2.39	3.13	3.09	2.29	3.95	4.85	2.40	3.28	4.55	2.47	3.10	4.50	2.01	2.26	3.45
		250	1.69	2.01	2.60	1.82	2.23	2.32	1.75	2.93	3.22	1.83	2.39	2.97	1.88	2.27	2.88	1.62	1.82	2.32
		500	1.50	1.70	1.90	1.58	1.83	1.95	1.61	2.39	2.55	1.64	2.01	2.32	1.63	1.89	2.07	1.37	1.49	1.78
2.5	0.75	100	1.64	2.31	3.38	2.40	3.25	3.17	1.70	3.65	4.54	1.67	3.41	4.82	2.31	3.02	4.64	1.55	2.03	2.83
		250	1.36	1.82	2.58	1.85	2.37	2.21	1.39	2.67	3.17	1.38	2.41	3.17	1.74	2.15	2.94	1.30	1.72	2.28
		500	1.26	1.49	2.14	1.59	1.99	1.66	1.21	2.16	2.49	1.20	2.03	2.38	1.53	1.75	2.30	1.22	1.54	1.97
5	0.25	100	2.77	3.38	5.12	2.87	3.57	3.62	3.12	4.95	9.03	3.14	3.65	8.15	3.07	3.81	6.00	2.54	2.85	3.90
		250	2.10	2.48	3.42	2.18	2.62	2.72	2.38	3.59	5.68	2.28	2.60	4.90	2.37	2.77	3.64	1.93	2.16	2.64
		500	1.76	2.06	2.72	1.83	2.15	2.22	2.00	2.84	3.95	1.92	2.17	3.41	1.93	2.24	2.55	1.65	1.80	2.19
5	0.5	100	2.54	3.15	5.29	2.94	3.65	3.67	2.88	4.65	8.10	3.09	3.60	7.27	3.07	3.78	6.40	2.33	2.72	4.55
		250	1.96	2.30	3.44	2.21	2.63	2.65	2.16	3.39	5.00	2.33	2.58	4.35	2.27	2.67	3.85	1.79	2.02	2.90
		500	1.64	1.93	2.63	1.86	2.14	2.04	1.85	2.72	3.50	1.95	2.12	3.05	1.88	2.13	2.87	1.58	1.76	2.22
5	0.75	100	2.05	2.60	4.80	2.97	3.70	3.73	2.20	4.09	6.57	2.27	3.36	5.91	2.80	3.57	6.58	1.89	2.36	4.26
		250	1.65	2.03	3.09	2.23	2.63	2.74	1.76	3.00	4.00	1.79	2.48	3.71	2.09	2.49	3.84	1.57	1.80	2.82
		500	1.46	1.78	2.43	1.82	2.10	2.11	1.55	2.45	3.01	1.58	2.06	2.67	1.74	2.02	2.83	1.40	1.58	2.13
10	0.25	100	3.45	4.14	6.19	3.55	4.25	4.23	4.05	6.15	15.77	3.56	4.04	18.20	3.97	4.70	8.39	3.18	3.51	5.48
		250	2.59	2.98	3.92	2.67	3.11	3.00	2.92	4.37	9.23	2.63	2.94	11.65	2.88	3.33	4.91	2.36	2.58	3.47
		500	2.11	2.39	2.98	2.21	2.48	2.45	2.38	3.38	6.17	2.13	2.34	8.37	2.35	2.68	3.55	1.95	2.11	2.54
10	0.5	100	3.12	3.82	7.72	3.74	4.49	4.50	3.75	5.76	13.68	3.63	4.05	13.00	3.88	4.66	9.60	2.86	3.29	6.48
		250	2.31	2.77	4.69	2.68	3.12	3.16	2.73	4.09	7.96	2.63	2.89	7.47	2.80	3.26	5.38	2.14	2.42	4.13
		500	1.93	2.23	3.36	2.19	2.47	2.49	2.18	3.18	5.30	2.11	2.33	5.05	2.26	2.56	3.76	1.76	1.97	3.04
10	0.75	100	2.50	3.16	6.90	3.80	4.53	4.68	2.92	4.84	9.98	3.04	3.77	8.66	3.69	4.57	10.22	2.33	2.84	5.86
		250	1.92	2.37	4.13	2.67	3.04	3.19	2.14	3.50	5.74	2.37	2.71	5.12	2.55	3.02	5.44	1.80	2.15	3.63
		500	1.66	1.97	3.04	2.14	2.43	2.53	1.82	2.78	4.07	1.99	2.19	3.60	2.05	2.33	3.73	1.56	1.79	2.76