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

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	\(\hat {\alpha }\)	0.0065	0.0041	0.0014	0.0011	0.0075	0.0051	0.0024	0.0019
	\(\hat {\rho }\)	0.9942	0.9840	0.9724	0.9589	0.9976	0.9939	0.9893	0.9833
	\(\hat {s}\)	0.0170	0.0157	0.0153	0.0138	0.0106	0.0078	0.0077	0.0074
α= 0.05	\(\hat {\alpha }\)	0.0256	0.0197	0.0133	0.0101	0.0285	0.0229	0.0184	0.0127
	\(\hat {\rho }\)	0.9836	0.9710	0.9592	0.9471	0.9925	0.9879	0.9830	0.9772
	\(\hat {s}\)	0.0315	0.0240	0.0201	0.0163	0.0205	0.0133	0.0111	0.0093
α= 0.1	\(\hat {\alpha }\)	0.0507	0.0401	0.0326	0.0228	0.0549	0.0467	0.0384	0.0285
	\(\hat {\rho }\)	0.9741	0.9618	0.9510	0.9403	0.9876	0.9834	0.9787	0.9735
	\(\hat {s}\)	0.0411	0.0288	0.0226	0.0176	0.0275	0.0169	0.0131	0.0104
α= 0.2	\(\hat {\alpha }\)	0.1045	0.0877	0.0754	0.0613	0.1122	0.1005	0.0846	0.0706
	\(\hat {\rho }\)	0.9588	0.9489	0.9402	0.9317	0.9790	0.9765	0.9728	0.9687
	\(\hat {s}\)	0.0536	0.0346	0.0255	0.0190	0.0372	0.0215	0.0156	0.0117
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0083	0.0063	0.0036	0.0027	0.0084	0.0064	0.0033	0.0035
	\(\hat {\rho }\)	0.9995	0.9992	0.9986	0.9978	0.9997	0.9997	0.9995	0.9991
	\(\hat {s}\)	0.0050	0.0017	0.0015	0.0015	0.0039	0.0010	0.0007	0.0007
α= 0.05	\(\hat {\alpha }\)	0.0312	0.0267	0.0227	0.0167	0.0320	0.0280	0.0229	0.0178
	\(\hat {\rho }\)	0.9983	0.9981	0.9975	0.9967	0.9990	0.9991	0.9989	0.9985
	\(\hat {s}\)	0.0099	0.0036	0.0026	0.0022	0.0079	0.0022	0.0014	0.0011
α= 0.1	\(\hat {\alpha }\)	0.0604	0.0544	0.0478	0.0376	0.0610	0.0572	0.0490	0.0394
	\(\hat {\rho }\)	0.9969	0.9972	0.9967	0.9958	0.9981	0.9986	0.9985	0.9981
	\(\hat {s}\)	0.0136	0.0051	0.0034	0.0027	0.0108	0.0032	0.0019	0.0014
α= 0.2	\(\hat {\alpha }\)	0.1203	0.1120	0.0961	0.0823	0.1218	0.1151	0.0985	0.0866
	\(\hat {\rho }\)	0.9943	0.9955	0.9953	0.9947	0.9964	0.9977	0.9977	0.9975
	\(\hat {s}\)	0.0190	0.0072	0.0045	0.0033	0.0152	0.0047	0.0027	0.0018

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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