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 42 Performance of nonparametric tolerance intervals when the true underlying distribution is 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	\(\hat {\alpha }\)	0.2000	0.1393	0.0230	0.0095	0.1394	0.2251	0.1331	0.0284
	\(\hat {\rho }\)	0.9424	0.9485	0.9662	0.9592	0.9778	0.9680	0.9745	0.9830
	\(\hat {s}\)	0.0784	0.0458	0.0256	0.0195	0.0504	0.0385	0.0233	0.0133
α= 0.05	\(\hat {\alpha }\)	0.2078	0.1520	0.0389	0.0248	0.1448	0.2351	0.1456	0.0442
	\(\hat {\rho }\)	0.9401	0.9457	0.9584	0.9487	0.9769	0.9668	0.9731	0.9792
	\(\hat {s}\)	0.0797	0.0466	0.0274	0.0210	0.0514	0.0391	0.0237	0.0142
α= 0.1	\(\hat {\alpha }\)	0.2213	0.1713	0.0964	0.1088	0.1530	0.2519	0.1655	0.0915
	\(\hat {\rho }\)	0.9369	0.9417	0.9423	0.9314	0.9756	0.9651	0.9711	0.9722
	\(\hat {s}\)	0.0813	0.0477	0.0311	0.0244	0.0528	0.0398	0.0243	0.0158
α= 0.2	\(\hat {\alpha }\)	0.2521	0.2206	0.1201	0.1334	0.1713	0.2862	0.2263	0.1224
	\(\hat {\rho }\)	0.9296	0.9317	0.9387	0.9282	0.9726	0.9611	0.9655	0.9693
	\(\hat {s}\)	0.0850	0.0500	0.0318	0.0248	0.0559	0.0415	0.0261	0.0162
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0178	0.0816	0.1650	0.2303	0.0055	0.0344	0.0827	0.1689
	\(\hat {\rho }\)	0.9989	0.9964	0.9946	0.9933	0.9997	0.9989	0.9983	0.9972
	\(\hat {s}\)	0.0113	0.0143	0.0117	0.0092	0.0054	0.0074	0.0065	0.0063
α= 0.05	\(\hat {\alpha }\)	0.0184	0.0854	0.1717	0.2405	0.0058	0.0356	0.0850	0.1758
	\(\hat {\rho }\)	0.9988	0.9962	0.9944	0.9931	0.9997	0.9989	0.9983	0.9971
	\(\hat {s}\)	0.0116	0.0146	0.0119	0.0094	0.0056	0.0075	0.0066	0.0064
α= 0.1	\(\hat {\alpha }\)	0.0197	0.0911	0.1806	0.2564	0.0062	0.0375	0.0900	0.1835
	\(\hat {\rho }\)	0.9987	0.9960	0.9941	0.9927	0.9997	0.9988	0.9981	0.9969
	\(\hat {s}\)	0.0119	0.0150	0.0122	0.0096	0.0057	0.0077	0.0068	0.0066
α= 0.2	\(\hat {\alpha }\)	0.0219	0.1013	0.2013	0.2840	0.0070	0.0417	0.1015	0.2076
	\(\hat {\rho }\)	0.9986	0.9956	0.9932	0.9920	0.9996	0.9987	0.9978	0.9966
	\(\hat {s}\)	0.0127	0.0158	0.0134	0.0098	0.0061	0.0082	0.0077	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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