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 14 Performance of parametric tolerance intervals based on the proposed model selection approach for symmetric distributions when the true underlying distribution is normal

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	\(\hat {\alpha }\)	0.0177	0.0175	0.0148	0.0122	0.0154	0.0161	0.0131	0.0109
	\(\hat {\rho }\)	0.9920	0.9789	0.9637	0.9488	0.9970	0.9926	0.9868	0.9798
	\(\hat {s}\)	0.0262	0.0244	0.0218	0.0176	0.0139	0.0121	0.0111	0.0096
α= 0.05	\(\hat {\alpha }\)	0.0504	0.0557	0.0513	0.0472	0.0471	0.0513	0.0464	0.0431
	\(\hat {\rho }\)	0.9788	0.9621	0.9481	0.9359	0.9912	0.9851	0.9790	0.9728
	\(\hat {s}\)	0.0415	0.0330	0.0262	0.0198	0.0241	0.0183	0.0149	0.0117
α= 0.1	\(\hat {\alpha }\)	0.0882	0.0977	0.0948	0.0916	0.0811	0.0928	0.0882	0.0838
	\(\hat {\rho }\)	0.9659	0.9503	0.9385	0.9286	0.9851	0.9792	0.9738	0.9686
	\(\hat {s}\)	0.0522	0.0376	0.0285	0.0209	0.0321	0.0222	0.0171	0.0129
α= 0.2	\(\hat {\alpha }\)	0.1730	0.1861	0.1874	0.1793	0.1592	0.1757	0.1759	0.1676
	\(\hat {\rho }\)	0.9438	0.9339	0.9260	0.9193	0.9734	0.9703	0.9667	0.9630
	\(\hat {s}\)	0.0661	0.0432	0.0312	0.0222	0.0437	0.0274	0.0199	0.0144
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0125	0.0120	0.0100	0.0079	0.0114	0.0104	0.0096	0.0076
	\(\hat {\rho }\)	0.9995	0.9992	0.9987	0.9977	0.9997	0.9997	0.9995	0.9991
	\(\hat {s}\)	0.0041	0.0025	0.0020	0.0020	0.0025	0.0013	0.0010	0.0010
α= 0.05	\(\hat {\alpha }\)	0.0424	0.0440	0.0398	0.0373	0.0406	0.0421	0.0387	0.0370
	\(\hat {\rho }\)	0.9983	0.9980	0.9973	0.9962	0.9990	0.9991	0.9988	0.9984
	\(\hat {s}\)	0.0085	0.0047	0.0034	0.0029	0.0058	0.0027	0.0018	0.0015
α= 0.1	\(\hat {\alpha }\)	0.0755	0.0809	0.0783	0.0759	0.0735	0.0774	0.0762	0.0757
	\(\hat {\rho }\)	0.9967	0.9967	0.9962	0.9953	0.9981	0.9984	0.9983	0.9979
	\(\hat {s}\)	0.0125	0.0063	0.0044	0.0035	0.0088	0.0039	0.0025	0.0019
α= 0.2	\(\hat {\alpha }\)	0.1460	0.1593	0.1607	0.1567	0.1432	0.1558	0.1576	0.1550
	\(\hat {\rho }\)	0.9934	0.9946	0.9944	0.9938	0.9960	0.9972	0.9973	0.9971
	\(\hat {s}\)	0.0190	0.0090	0.0059	0.0043	0.0140	0.0057	0.0035	0.0025

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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