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 39 Performance of nonparametric tolerance intervals when the true underlying distribution is 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	\(\hat {\alpha }\)	0.1885	0.1331	0.0237	0.0077	0.1246	0.2122	0.1316	0.0244
	\(\hat {\rho }\)	0.9471	0.9510	0.9668	0.9594	0.9803	0.9704	0.9754	0.9832
	\(\hat {s}\)	0.0763	0.0457	0.0254	0.0191	0.0494	0.0383	0.0232	0.0132
α= 0.05	\(\hat {\alpha }\)	0.1998	0.1447	0.0380	0.0208	0.1298	0.2221	0.1445	0.0421
	\(\hat {\rho }\)	0.9449	0.9481	0.9582	0.9495	0.9795	0.9692	0.9740	0.9791
	\(\hat {s}\)	0.0776	0.0465	0.0271	0.0207	0.0504	0.0389	0.0236	0.0141
α= 0.1	\(\hat {\alpha }\)	0.2129	0.1624	0.1027	0.1011	0.1365	0.2347	0.1622	0.0961
	\(\hat {\rho }\)	0.9418	0.9440	0.9411	0.9316	0.9783	0.9676	0.9719	0.9717
	\(\hat {s}\)	0.0793	0.0475	0.0309	0.0241	0.0517	0.0397	0.0242	0.0158
α= 0.2	\(\hat {\alpha }\)	0.2424	0.2123	0.1126	0.1220	0.1528	0.2639	0.2133	0.1179
	\(\hat {\rho }\)	0.9347	0.9335	0.9395	0.9290	0.9756	0.9637	0.9667	0.9694
	\(\hat {s}\)	0.0830	0.0497	0.0314	0.0245	0.0545	0.0414	0.0254	0.0161
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0154	0.0717	0.1503	0.2138	0.0046	0.0242	0.0737	0.1538
	\(\hat {\rho }\)	0.9990	0.9969	0.9950	0.9937	0.9998	0.9991	0.9983	0.9974
	\(\hat {s}\)	0.0109	0.0134	0.0119	0.0091	0.0050	0.0075	0.0071	0.0061
α= 0.05	\(\hat {\alpha }\)	0.0162	0.0753	0.1556	0.2239	0.0048	0.0253	0.0767	0.1583
	\(\hat {\rho }\)	0.9989	0.9968	0.9948	0.9935	0.9998	0.9991	0.9983	0.9973
	\(\hat {s}\)	0.0112	0.0136	0.0121	0.0093	0.0051	0.0076	0.0073	0.0062
α= 0.1	\(\hat {\alpha }\)	0.0168	0.0802	0.1647	0.2382	0.0054	0.0265	0.0801	0.1678
	\(\hat {\rho }\)	0.9989	0.9966	0.9945	0.9932	0.9997	0.9991	0.9982	0.9972
	\(\hat {s}\)	0.0115	0.0140	0.0124	0.0095	0.0053	0.0078	0.0074	0.0064
α= 0.2	\(\hat {\alpha }\)	0.0189	0.0887	0.1824	0.2728	0.0063	0.0303	0.0902	0.1849
	\(\hat {\rho }\)	0.9987	0.9962	0.9940	0.9924	0.9997	0.9989	0.9980	0.9969
	\(\hat {s}\)	0.0123	0.0148	0.0128	0.0098	0.0056	0.0082	0.0076	0.0067

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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