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

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	\(\hat {\alpha }\)	0.0422	0.0370	0.0286	0.0228	0.0502	0.0667	0.0582	0.0454
	\(\hat {\rho }\)	0.9804	0.9722	0.9619	0.9500	0.9897	0.9860	0.9822	0.9773
	\(\hat {s}\)	0.0512	0.0318	0.0252	0.0189	0.0426	0.0215	0.0170	0.0129
α= 0.05	\(\hat {\alpha }\)	0.1014	0.0891	0.0703	0.0490	0.1228	0.1332	0.1120	0.0877
	\(\hat {\rho }\)	0.9628	0.9552	0.9469	0.9377	0.9798	0.9768	0.9741	0.9705
	\(\hat {s}\)	0.0528	0.0383	0.0285	0.0205	0.0370	0.0271	0.0199	0.0143
α= 0.1	\(\hat {\alpha }\)	0.1596	0.1331	0.1067	0.0797	0.1851	0.1841	0.1574	0.1209
	\(\hat {\rho }\)	0.9483	0.9444	0.9383	0.9310	0.9712	0.9707	0.9692	0.9667
	\(\hat {s}\)	0.0617	0.0419	0.0302	0.0213	0.0448	0.0303	0.0215	0.0150
α= 0.2	\(\hat {\alpha }\)	0.2569	0.2130	0.1774	0.1433	0.2860	0.2672	0.2303	0.1825
	\(\hat {\rho }\)	0.9273	0.9304	0.9277	0.9230	0.9579	0.9623	0.9629	0.9619
	\(\hat {s}\)	0.0727	0.0461	0.0323	0.0223	0.0550	0.0344	0.0235	0.0159
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0864	0.1367	0.1279	0.0970	0.1032	0.1664	0.1496	0.1076
	\(\hat {\rho }\)	0.9959	0.9958	0.9960	0.9959	0.9969	0.9972	0.9976	0.9979
	\(\hat {s}\)	0.0369	0.0093	0.0070	0.0050	0.0360	0.0068	0.0049	0.0033
α= 0.05	\(\hat {\alpha }\)	0.1904	0.2190	0.1853	0.1368	0.2157	0.2467	0.2003	0.1463
	\(\hat {\rho }\)	0.9922	0.9929	0.9939	0.9942	0.9943	0.9953	0.9964	0.9970
	\(\hat {s}\)	0.0200	0.0127	0.0087	0.0058	0.0162	0.0095	0.0063	0.0039
α= 0.1	\(\hat {\alpha }\)	0.2624	0.2722	0.2245	0.1677	0.2955	0.2946	0.2377	0.1778
	\(\hat {\rho }\)	0.9885	0.9909	0.9925	0.9932	0.9916	0.9939	0.9956	0.9964
	\(\hat {s}\)	0.0252	0.0148	0.0098	0.0062	0.0206	0.0113	0.0072	0.0043
α= 0.2	\(\hat {\alpha }\)	0.3705	0.3432	0.2859	0.2263	0.3989	0.3599	0.2959	0.2345
	\(\hat {\rho }\)	0.9825	0.9879	0.9906	0.9919	0.9870	0.9919	0.9944	0.9957
	\(\hat {s}\)	0.0325	0.0176	0.0111	0.0068	0.0269	0.0136	0.0083	0.0048

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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