Tolerance intervals in statistical software and robustness under model misspecification

Cho, Kyung Serk; Tony Ng, Hon Keung

doi:10.1186/s40488-021-00123-2

Journal of Statistical Distributions and Applications

Table 37 Performance of tolerance intervals based on gamma distribution when the true distribution is lognormal (G: Gamma; F: Lognormal)

From: Tolerance intervals in statistical software and robustness under model misspecification

		n = 10	n = 25	n = 50	n = 100	n = 10	n = 25	n = 50	n = 100
		ρ= 0.9				ρ= 0.95
α = 0.01	\(\hat {\alpha }\)	0.0159	0.0155	0.0142	0.0091	0.0183	0.0185	0.0180	0.0133
	\(\hat {\rho }\)	0.9879	0.9743	0.9616	0.9490	0.9947	0.9893	0.9836	0.9775
	\(\hat {s}\)	0.0260	0.0231	0.0207	0.0172	0.0159	0.0126	0.0114	0.0098
α= 0.05	\(\hat {\alpha }\)	0.0775	0.0642	0.0562	0.0488	0.0831	0.0742	0.0669	0.0619
	\(\hat {\rho }\)	0.9680	0.9561	0.9460	0.9366	0.9843	0.9800	0.9754	0.9706
	\(\hat {s}\)	0.0466	0.0326	0.0257	0.0197	0.0305	0.0196	0.0153	0.0119
α= 0.1	\(\hat {\alpha }\)	0.1407	0.1221	0.1004	0.0933	0.1472	0.1344	0.1176	0.1108
	\(\hat {\rho }\)	0.9514	0.9438	0.9364	0.9293	0.9747	0.9732	0.9700	0.9664
	\(\hat {s}\)	0.0592	0.0378	0.0283	0.0210	0.0405	0.0238	0.0175	0.0130
α= 0.2	\(\hat {\alpha }\)	0.2640	0.2238	0.1971	0.1778	0.2716	0.2395	0.2174	0.1996
	\(\hat {\rho }\)	0.9254	0.9266	0.9238	0.9201	0.9586	0.9630	0.9625	0.9609
	\(\hat {s}\)	0.0744	0.0440	0.0313	0.0225	0.0541	0.0293	0.0203	0.0145
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0450	0.0698	0.0923	0.1367	0.0566	0.1061	0.1650	0.2684
α= 0.01	\(\hat {\alpha }\)	0.0253	0.0330	0.0446	0.0569	0.0295	0.0434	0.0630	0.1008
	\(\hat {\rho }\)	0.9988	0.9979	0.9967	0.9952	0.9993	0.9988	0.9982	0.9973
	\(\hat {s}\)	0.0069	0.0039	0.0034	0.0029	0.0053	0.0025	0.0021	0.0018
α= 0.05	\(\hat {\alpha }\)	0.1027	0.1190	0.1300	0.1615	0.1133	0.1462	0.1766	0.2383
	\(\hat {\rho }\)	0.9957	0.9954	0.9945	0.9934	0.9972	0.9973	0.9968	0.9961
	\(\hat {s}\)	0.0142	0.0068	0.0050	0.0038	0.0110	0.0046	0.0033	0.0024
α= 0.1	\(\hat {\alpha }\)	0.1805	0.1952	0.2063	0.2418	0.1970	0.2335	0.2743	0.3458
	\(\hat {\rho }\)	0.9924	0.9934	0.9929	0.9922	0.9950	0.9960	0.9958	0.9954
	\(\hat {s}\)	0.0198	0.0089	0.0060	0.0043	0.0155	0.0061	0.0040	0.0028
α = 0.2	\(\hat {\alpha }\)	0.3116	0.3176	0.3402	0.3725	0.3352	0.3686	0.4173	0.4913
	\(\hat {\rho }\)	0.9861	0.9902	0.9907	0.9905	0.9906	0.9939	0.9944	0.9943
	\(\hat {s}\)	0.0283	0.0119	0.0075	0.0051	0.0224	0.0084	0.0051	0.0034

The bolded values are \({\hat \alpha }\) within \(\pm 2 \sqrt {\alpha (1-\alpha)/M}\) and \({\hat \rho }\) within \(\pm 2 \sqrt {\rho (1-\rho)/M}\)

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