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 15 Performance of parametric tolerance intervals based on the model selection approach for symmetric distributions when the true 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	\(\hat {\alpha }\)	0.1671	0.0829	0.0565	0.0224	0.2399	0.1721	0.0847	0.0236
	\(\hat {\rho }\)	0.9291	0.9326	0.9291	0.9249	0.9507	0.9618	0.9629	0.9623
	\(\hat {s}\)	0.1123	0.0236	0.0169	0.0111	0.1244	0.0199	0.0135	0.0065
α= 0.05	\(\hat {\alpha }\)	0.2737	0.1658	0.1055	0.0567	0.3868	0.2383	0.1221	0.0582
	\(\hat {\rho }\)	0.9124	0.9231	0.9226	0.9197	0.9407	0.9561	0.9594	0.9597
	\(\hat {s}\)	0.0964	0.0282	0.0190	0.0119	0.0972	0.0246	0.0155	0.0070
α= 0.1	\(\hat {\alpha }\)	0.3461	0.2176	0.1430	0.0863	0.4543	0.2769	0.1545	0.0882
	\(\hat {\rho }\)	0.9044	0.9177	0.9188	0.9168	0.9355	0.9528	0.9574	0.9583
	\(\hat {s}\)	0.0723	0.0307	0.0201	0.0123	0.0740	0.0271	0.0165	0.0073
α= 0.2	\(\hat {\alpha }\)	0.4319	0.2836	0.2003	0.1414	0.5245	0.3257	0.2083	0.1424
	\(\hat {\rho }\)	0.8916	0.9109	0.9140	0.9132	0.9273	0.9488	0.9549	0.9564
	\(\hat {s}\)	0.0714	0.0337	0.0214	0.0128	0.0638	0.0299	0.0176	0.0076
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.4342	0.2143	0.0862	0.0238	0.4681	0.2143	0.0862	0.0238
	\(\hat {\rho }\)	0.9698	0.9866	0.9904	0.9923	0.9728	0.9902	0.9941	0.9960
	\(\hat {s}\)	0.1388	0.0174	0.0113	0.0037	0.1414	0.0162	0.0104	0.0034
α= 0.05	\(\hat {\alpha }\)	0.5075	0.2529	0.1227	0.0585	0.5107	0.2529	0.1227	0.0585
	\(\hat {\rho }\)	0.9665	0.9841	0.9894	0.9917	0.9710	0.9883	0.9934	0.9958
	\(\hat {s}\)	0.1011	0.0213	0.0129	0.0041	0.1006	0.0197	0.0118	0.0036
α= 0.1	\(\hat {\alpha }\)	0.5329	0.2836	0.1547	0.0884	0.5349	0.2836	0.1547	0.0884
	\(\hat {\rho }\)	0.9650	0.9828	0.9889	0.9914	0.9701	0.9872	0.9931	0.9956
	\(\hat {s}\)	0.0698	0.0233	0.0137	0.0042	0.0683	0.0215	0.0125	0.0038
α= 0.2	\(\hat {\alpha }\)	0.5699	0.3286	0.2087	0.1426	0.5708	0.3286	0.2087	0.1426
	\(\hat {\rho }\)	0.9607	0.9812	0.9882	0.9911	0.9666	0.9860	0.9926	0.9954
	\(\hat {s}\)	0.0561	0.0256	0.0146	0.0044	0.0534	0.0236	0.0134	0.0039

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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