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 40 Performance of nonparametric tolerance intervals when the true underlying distribution is Cauchy

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	\(\hat {\alpha }\)	0.3251	0.1595	0.0279	0.0073	0.3140	0.3285	0.1717	0.0268
	\(\hat {\rho }\)	0.9108	0.9411	0.9641	0.9598	0.9513	0.9558	0.9698	0.9822
	\(\hat {s}\)	0.0784	0.0444	0.0260	0.0191	0.0565	0.0381	0.0239	0.0131
α= 0.05	\(\hat {\alpha }\)	0.3365	0.1703	0.0386	0.0318	0.3256	0.3412	0.1824	0.0396
	\(\hat {\rho }\)	0.9086	0.9391	0.9594	0.9466	0.9500	0.9548	0.9688	0.9798
	\(\hat {s}\)	0.0794	0.0452	0.0277	0.0216	0.0573	0.0386	0.0243	0.0140
α= 0.1	\(\hat {\alpha }\)	0.3520	0.1857	0.0744	0.0922	0.3399	0.3557	0.1974	0.0652
	\(\hat {\rho }\)	0.9057	0.9364	0.9498	0.9336	0.9482	0.9534	0.9674	0.9759
	\(\hat {s}\)	0.0807	0.0462	0.0310	0.0243	0.0584	0.0392	0.0248	0.0153
α= 0.2	\(\hat {\alpha }\)	0.3893	0.2240	0.1536	0.1522	0.3700	0.3880	0.2371	0.1415
	\(\hat {\rho }\)	0.8991	0.9298	0.9342	0.9259	0.9442	0.9503	0.9642	0.9675
	\(\hat {s}\)	0.0836	0.0485	0.0338	0.0256	0.0609	0.0406	0.0260	0.0168
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.1119	0.2781	0.3779	0.3642	0.0591	0.1796	0.3029	0.3742
	\(\hat {\rho }\)	0.9943	0.9898	0.9882	0.9899	0.9982	0.9961	0.9944	0.9941
	\(\hat {s}\)	0.0158	0.0162	0.0143	0.0097	0.0075	0.0092	0.0092	0.0071
α= 0.05	\(\hat {\alpha }\)	0.1159	0.2880	0.3884	0.3763	0.0616	0.1849	0.3114	0.3854
	\(\hat {\rho }\)	0.9941	0.9895	0.9879	0.9897	0.9981	0.9959	0.9942	0.9940
	\(\hat {s}\)	0.0161	0.0165	0.0145	0.0098	0.0077	0.0093	0.0094	0.0072
α= 0.1	\(\hat {\alpha }\)	0.1228	0.3005	0.4016	0.3916	0.0647	0.1949	0.3221	0.4011
	\(\hat {\rho }\)	0.9938	0.9891	0.9875	0.9894	0.9980	0.9958	0.9940	0.9938
	\(\hat {s}\)	0.0165	0.0169	0.0148	0.0099	0.0079	0.0096	0.0096	0.0074
α= 0.2	\(\hat {\alpha }\)	0.1375	0.3299	0.4294	0.4232	0.0730	0.2176	0.3418	0.4300
	\(\hat {\rho }\)	0.9932	0.9882	0.9868	0.9886	0.9978	0.9954	0.9937	0.9933
	\(\hat {s}\)	0.0176	0.0177	0.0151	0.0102	0.0085	0.0101	0.0100	0.0076

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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