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 3 Performance of tolerance intervals based on normal distribution when the true distribution is normal (F=G: Normal)

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	\(\hat {\alpha }\)	0.0112	0.0109	0.0107	0.0115	0.0108	0.0107	0.0106	0.0115
	\(\hat {\rho }\)	0.9926	0.9784	0.9629	0.9480	0.9973	0.9926	0.9862	0.9789
	\(\hat {s}\)	0.0206	0.0209	0.0200	0.0171	0.0111	0.0103	0.0104	0.0095
α= 0.05	\(\hat {\alpha }\)	0.0519	0.0506	0.0515	0.0546	0.0514	0.0501	0.0513	0.0546
	\(\hat {\rho }\)	0.9765	0.9602	0.9465	0.9347	0.9898	0.9839	0.9775	0.9714
	\(\hat {s}\)	0.0403	0.0306	0.0250	0.0196	0.0248	0.0173	0.0145	0.0117
α= 0.1	\(\hat {\alpha }\)	0.0995	0.0991	0.1010	0.1044	0.0983	0.0984	0.1009	0.1043
	\(\hat {\rho }\)	0.9620	0.9478	0.9365	0.9271	0.9822	0.9773	0.9719	0.9668
	\(\hat {s}\)	0.0527	0.0358	0.0276	0.0208	0.0345	0.0217	0.0168	0.0129
α= 0.2	\(\hat {\alpha }\)	0.1960	0.1951	0.2049	0.2011	0.1936	0.1940	0.2047	0.2009
	\(\hat {\rho }\)	0.9385	0.9306	0.9234	0.9174	0.9684	0.9672	0.9639	0.9608
	\(\hat {s}\)	0.0679	0.0420	0.0306	0.0223	0.0477	0.0272	0.0197	0.0144
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0105	0.0103	0.0106	0.0115	0.0104	0.0103	0.0105	0.0115
	\(\hat {\rho }\)	0.9996	0.9992	0.9985	0.9973	0.9998	0.9997	0.9994	0.9989
	\(\hat {s}\)	0.0035	0.0022	0.0021	0.0021	0.0023	0.0012	0.0011	0.0010
α= 0.05	\(\hat {\alpha }\)	0.0492	0.0495	0.0509	0.0544	0.0486	0.0491	0.0509	0.0544
	\(\hat {\rho }\)	0.9979	0.9977	0.9968	0.9956	0.9988	0.9989	0.9985	0.9980
	\(\hat {s}\)	0.0096	0.0047	0.0037	0.0030	0.0068	0.0028	0.0020	0.0016
α= 0.1	\(\hat {\alpha }\)	0.0963	0.0977	0.1003	0.1040	0.0957	0.0970	0.1000	0.1038
	\(\hat {\rho }\)	0.9957	0.9961	0.9954	0.9945	0.9974	0.9981	0.9978	0.9974
	\(\hat {s}\)	0.0147	0.0066	0.0047	0.0036	0.0108	0.0041	0.0027	0.0020
α= 0.2	\(\hat {\alpha }\)	0.1905	0.1920	0.2040	0.2006	0.1895	0.1915	0.2036	0.2005
	\(\hat {\rho }\)	0.9912	0.9934	0.9933	0.9928	0.9945	0.9965	0.9967	0.9965
	\(\hat {s}\)	0.0227	0.0095	0.0062	0.0044	0.0172	0.0062	0.0037	0.0025

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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