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 6 Performance of tolerance intervals based on logistic distribution when the true distribution is logistic (F=G: Logistic)

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	\(\hat {\alpha }\)	0.0750	0.0139	0.0067	0.0021	0.0820	0.0182	0.0101	0.0037
	\(\hat {\rho }\)	0.9231	0.9842	0.9761	0.9647	0.9306	0.9927	0.9894	0.9842
	\(\hat {s}\)	0.3410	0.0231	0.0194	0.0160	0.3409	0.0132	0.0105	0.0087
α= 0.05	\(\hat {\alpha }\)	0.0980	0.0348	0.0235	0.0116	0.1114	0.0454	0.0321	0.0194
	\(\hat {\rho }\)	0.9657	0.9719	0.9633	0.9526	0.9789	0.9865	0.9830	0.9780
	\(\hat {s}\)	0.0914	0.0309	0.0242	0.0185	0.0801	0.0182	0.0137	0.0105
α= 0.1	\(\hat {\alpha }\)	0.1335	0.0589	0.0420	0.0268	0.1473	0.0745	0.0544	0.0363
	\(\hat {\rho }\)	0.9576	0.9631	0.9551	0.9453	0.9749	0.9819	0.9787	0.9742
	\(\hat {s}\)	0.0764	0.0351	0.0266	0.0197	0.0584	0.0212	0.0155	0.0114
α= 0.2	\(\hat {\alpha }\)	0.1893	0.1064	0.0809	0.0585	0.2051	0.1273	0.0993	0.0771
	\(\hat {\rho }\)	0.9430	0.9505	0.9440	0.9361	0.9663	0.9751	0.9728	0.9692
	\(\hat {s}\)	0.0799	0.0401	0.0293	0.0210	0.0580	0.0249	0.0176	0.0125
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.1019	0.0250	0.0162	0.0075	0.1038	0.0266	0.0183	0.0096
	\(\hat {\rho }\)	0.9329	0.9987	0.9983	0.9975	0.9342	0.9993	0.9992	0.9989
	\(\hat {s}\)	0.3502	0.0038	0.0025	0.0020	0.3502	0.0023	0.0013	0.0010
α= 0.05	\(\hat {\alpha }\)	0.1238	0.0587	0.0456	0.0304	0.1271	0.0628	0.0488	0.0329
	\(\hat {\rho }\)	0.9916	0.9974	0.9970	0.9962	0.9938	0.9987	0.9986	0.9982
	\(\hat {s}\)	0.0698	0.0055	0.0035	0.0026	0.0683	0.0034	0.0020	0.0014
α= 0.1	\(\hat {\alpha }\)	0.1630	0.0927	0.0733	0.0530	0.1668	0.0970	0.0778	0.0588
	\(\hat {\rho }\)	0.9915	0.9963	0.9961	0.9954	0.9944	0.9981	0.9981	0.9978
	\(\hat {s}\)	0.0378	0.0066	0.0042	0.0029	0.0340	0.0041	0.0024	0.0016
α= 0.2	\(\hat {\alpha }\)	0.2221	0.1518	0.1258	0.1063	0.2264	0.1571	0.1318	0.1138
	\(\hat {\rho }\)	0.9890	0.9947	0.9948	0.9943	0.9930	0.9972	0.9974	0.9972
	\(\hat {s}\)	0.0287	0.0081	0.0050	0.0034	0.0216	0.0050	0.0029	0.0019

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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