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 9 Performance of tolerance intervals based on logistic distribution when the true distribution is logistic (F=G: Lognormal)

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	\(\hat {\alpha }\)	0.0100	0.0100	0.0120	0.0084	0.0097	0.0100	0.0120	0.0084
	\(\hat {\rho }\)	0.9925	0.9783	0.9632	0.9485	0.9973	0.9926	0.9863	0.9792
	\(\hat {s}\)	0.0210	0.0210	0.0199	0.0168	0.0120	0.0104	0.0104	0.0093
α= 0.05	\(\hat {\alpha }\)	0.0529	0.0481	0.0481	0.0508	0.0524	0.0473	0.0480	0.0508
	\(\hat {\rho }\)	0.9762	0.9599	0.9469	0.9353	0.9897	0.9838	0.9778	0.9717
	\(\hat {s}\)	0.0404	0.0305	0.0248	0.0192	0.0251	0.0174	0.0144	0.0115
α= 0.1	\(\hat {\alpha }\)	0.1036	0.0991	0.0936	0.0975	0.1021	0.0985	0.0935	0.0975
	\(\hat {\rho }\)	0.9614	0.9475	0.9369	0.9277	0.9819	0.9771	0.9721	0.9672
	\(\hat {s}\)	0.0527	0.0357	0.0274	0.0205	0.0346	0.0217	0.0167	0.0126
α= 0.2	\(\hat {\alpha }\)	0.2040	0.1978	0.1932	0.1930	0.2015	0.1972	0.1927	0.1928
	\(\hat {\rho }\)	0.9376	0.9301	0.9239	0.9180	0.9679	0.9669	0.9642	0.9612
	\(\hat {s}\)	0.0679	0.0418	0.0303	0.0219	0.0477	0.0272	0.0195	0.0141
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0091	0.0098	0.0120	0.0084	0.0091	0.0098	0.0120	0.0083
	\(\hat {\rho }\)	0.9995	0.9992	0.9985	0.9974	0.9997	0.9997	0.9994	0.9989
	\(\hat {s}\)	0.0048	0.0023	0.0021	0.0020	0.0036	0.0013	0.0011	0.0010
α= 0.05	\(\hat {\alpha }\)	0.0501	0.0463	0.0476	0.0507	0.0495	0.0460	0.0475	0.0507
	\(\hat {\rho }\)	0.9978	0.9976	0.9968	0.9957	0.9987	0.9989	0.9986	0.9981
	\(\hat {s}\)	0.0106	0.0048	0.0037	0.0029	0.0079	0.0029	0.0021	0.0016
α= 0.1	\(\hat {\alpha }\)	0.1000	0.0975	0.0930	0.0973	0.0991	0.0971	0.0930	0.0973
	\(\hat {\rho }\)	0.9957	0.9961	0.9955	0.9946	0.9974	0.9981	0.9979	0.9975
	\(\hat {s}\)	0.0154	0.0067	0.0047	0.0035	0.0117	0.0042	0.0027	0.0019
α= 0.2	\(\hat {\alpha }\)	0.1971	0.1959	0.1915	0.1926	0.1952	0.1954	0.1913	0.1925
	\(\hat {\rho }\)	0.9911	0.9934	0.9934	0.9930	0.9945	0.9965	0.9967	0.9966
	\(\hat {s}\)	0.0231	0.0096	0.0062	0.0043	0.0177	0.0063	0.0038	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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