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 17 Performance of parametric tolerance intervals base on the proposed model selection approach for symmetric distributions when the true underlying true 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	\(\hat {\alpha }\)	0.0170	0.0152	0.0147	0.0120	0.0192	0.0243	0.0277	0.0311
	\(\hat {\rho }\)	0.9851	0.9769	0.9664	0.9565	0.9910	0.9893	0.9843	0.9795
	\(\hat {s}\)	0.0905	0.0248	0.0240	0.0211	0.0904	0.0142	0.0143	0.0132
α= 0.05	\(\hat {\alpha }\)	0.0563	0.0572	0.0531	0.0421	0.0640	0.0817	0.0871	0.0834
	\(\hat {\rho }\)	0.9730	0.9610	0.9524	0.9446	0.9866	0.9809	0.9769	0.9730
	\(\hat {s}\)	0.0464	0.0333	0.0284	0.0229	0.0332	0.0208	0.0181	0.0151
α= 0.1	\(\hat {\alpha }\)	0.1057	0.1021	0.0926	0.0728	0.1198	0.1377	0.1378	0.1318
	\(\hat {\rho }\)	0.9596	0.9503	0.9438	0.9376	0.9793	0.9750	0.9721	0.9691
	\(\hat {s}\)	0.0524	0.0380	0.0307	0.0238	0.0347	0.0246	0.0202	0.0161
α= 0.2	\(\hat {\alpha }\)	0.1985	0.1855	0.1628	0.1337	0.2153	0.2254	0.2173	0.2020
	\(\hat {\rho }\)	0.9381	0.9356	0.9326	0.9288	0.9665	0.9664	0.9656	0.9641
	\(\hat {s}\)	0.0658	0.0435	0.0333	0.0248	0.0459	0.0296	0.0227	0.0174
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0287	0.0583	0.0913	0.1335	0.0351	0.0791	0.1327	0.1925
	\(\hat {\rho }\)	0.9945	0.9975	0.9964	0.9955	0.9950	0.9985	0.9979	0.9974
	\(\hat {s}\)	0.0917	0.0046	0.0047	0.0046	0.0916	0.0030	0.0031	0.0031
α= 0.05	\(\hat {\alpha }\)	0.0997	0.1578	0.1963	0.2281	0.1161	0.1966	0.2492	0.2792
	\(\hat {\rho }\)	0.9959	0.9950	0.9943	0.9937	0.9973	0.9969	0.9966	0.9964
	\(\hat {s}\)	0.0232	0.0077	0.0067	0.0059	0.0219	0.0053	0.0046	0.0040
α= 0.1	\(\hat {\alpha }\)	0.1646	0.2323	0.2675	0.2800	0.1890	0.2783	0.3166	0.3197
	\(\hat {\rho }\)	0.9935	0.9931	0.9929	0.9927	0.9957	0.9957	0.9957	0.9957
	\(\hat {s}\)	0.0162	0.0098	0.0080	0.0066	0.0124	0.0069	0.0055	0.0046
α= 0.2	\(\hat {\alpha }\)	0.2820	0.3364	0.3490	0.3369	0.3128	0.3765	0.3910	0.3611
	\(\hat {\rho }\)	0.9887	0.9902	0.9909	0.9912	0.9922	0.9937	0.9944	0.9948
	\(\hat {s}\)	0.0230	0.0128	0.0096	0.0075	0.0179	0.0092	0.0068	0.0053

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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