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 41 Performance of nonparametric tolerance intervals 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	\(\hat {\alpha }\)	0.2286	0.1375	0.0263	0.0092	0.1594	0.2278	0.1412	0.0250
	\(\hat {\rho }\)	0.9364	0.9486	0.9657	0.9595	0.9744	0.9678	0.9739	0.9831
	\(\hat {s}\)	0.0803	0.0446	0.0259	0.0194	0.0545	0.0378	0.0236	0.0130
α= 0.05	\(\hat {\alpha }\)	0.2383	0.1511	0.0428	0.0257	0.1668	0.2396	0.1538	0.0408
	\(\hat {\rho }\)	0.9340	0.9459	0.9579	0.9488	0.9733	0.9665	0.9725	0.9792
	\(\hat {s}\)	0.0815	0.0454	0.0277	0.0211	0.0555	0.0384	0.0241	0.0139
α= 0.1	\(\hat {\alpha }\)	0.2516	0.1686	0.1009	0.1030	0.1756	0.2528	0.1719	0.0918
	\(\hat {\rho }\)	0.9307	0.9420	0.9419	0.9317	0.9719	0.9649	0.9705	0.9722
	\(\hat {s}\)	0.0830	0.0464	0.0311	0.0244	0.0568	0.0391	0.0246	0.0156
α= 0.2	\(\hat {\alpha }\)	0.2858	0.2204	0.1181	0.1287	0.1967	0.2857	0.2215	0.1203
	\(\hat {\rho }\)	0.9232	0.9323	0.9387	0.9284	0.9686	0.9609	0.9659	0.9691
	\(\hat {s}\)	0.0863	0.0486	0.0317	0.0248	0.0596	0.0406	0.0258	0.0162
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0229	0.0844	0.1703	0.2248	0.0081	0.0349	0.0878	0.1636
	\(\hat {\rho }\)	0.9984	0.9963	0.9943	0.9934	0.9995	0.9989	0.9981	0.9972
	\(\hat {s}\)	0.0149	0.0145	0.0124	0.0090	0.0082	0.0080	0.0074	0.0062
α= 0.05	\(\hat {\alpha }\)	0.0234	0.0875	0.1763	0.2363	0.0083	0.0361	0.0909	0.1691
	\(\hat {\rho }\)	0.9983	0.9961	0.9941	0.9932	0.9995	0.9989	0.9980	0.9971
	\(\hat {s}\)	0.0151	0.0147	0.0126	0.0092	0.0084	0.0081	0.0076	0.0063
α= 0.1	\(\hat {\alpha }\)	0.0239	0.0928	0.1860	0.2512	0.0089	0.0374	0.0954	0.1793
	\(\hat {\rho }\)	0.9982	0.9959	0.9938	0.9928	0.9995	0.9988	0.9979	0.9969
	\(\hat {s}\)	0.0155	0.0151	0.0129	0.0094	0.0086	0.0083	0.0078	0.0064
α= 0.2	\(\hat {\alpha }\)	0.0268	0.1030	0.2048	0.2870	0.0100	0.0423	0.1029	0.2050
	\(\hat {\rho }\)	0.9980	0.9954	0.9931	0.9919	0.9994	0.9986	0.9977	0.9965
	\(\hat {s}\)	0.0163	0.0160	0.0134	0.0099	0.0090	0.0087	0.0078	0.0069

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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