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 26 Performance of tolerance intervals based on logistic distribution when the true distribution is normal (G: Logistic; F: 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	\(\hat {\alpha }\)	0.1173	0.0501	0.0237	0.0128	0.1119	0.0430	0.0160	0.0063
	\(\hat {\rho }\)	0.9630	0.9697	0.9640	0.9539	0.9809	0.9886	0.9878	0.9844
	\(\hat {s}\)	0.0673	0.0348	0.0248	0.0193	0.0455	0.0186	0.0120	0.0092
α= 0.05	\(\hat {\alpha }\)	0.1768	0.0909	0.0547	0.0344	0.1670	0.0760	0.0355	0.0165
	\(\hat {\rho }\)	0.9468	0.9562	0.9520	0.9436	0.9723	0.9827	0.9827	0.9799
	\(\hat {s}\)	0.0776	0.0409	0.0281	0.0210	0.0533	0.0229	0.0145	0.0106
α= 0.1	\(\hat {\alpha }\)	0.2180	0.1257	0.0801	0.0585	0.2046	0.1042	0.0547	0.0280
	\(\hat {\rho }\)	0.9360	0.9478	0.9449	0.9378	0.9665	0.9788	0.9795	0.9772
	\(\hat {s}\)	0.0829	0.0439	0.0298	0.0218	0.0576	0.0252	0.0158	0.0114
α= 0.2	\(\hat {\alpha }\)	0.2825	0.1841	0.1238	0.0961	0.2582	0.1460	0.0825	0.0452
	\(\hat {\rho }\)	0.9212	0.9366	0.9359	0.9307	0.9584	0.9734	0.9753	0.9739
	\(\hat {s}\)	0.0887	0.0472	0.0315	0.0226	0.0627	0.0280	0.0174	0.0122
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0874	0.0227	0.0045	0.0005	0.0780	0.0159	0.0033	0.0001
	\(\hat {\rho }\)	0.9955	0.9989	0.9993	0.9992	0.9975	0.9996	0.9998	0.9998
	\(\hat {s}\)	0.0189	0.0043	0.0016	0.0009	0.0133	0.0024	0.0006	0.0003
α= 0.05	\(\hat {\alpha }\)	0.1286	0.0401	0.0094	0.0011	0.1106	0.0280	0.0048	0.0003
	\(\hat {\rho }\)	0.9935	0.9982	0.9988	0.9988	0.9964	0.9993	0.9997	0.9997
	\(\hat {s}\)	0.0224	0.0054	0.0021	0.0012	0.0157	0.0030	0.0008	0.0004
α= 0.1	\(\hat {\alpha }\)	0.1557	0.0533	0.0139	0.0014	0.1359	0.0390	0.0068	0.0005
	\(\hat {\rho }\)	0.9921	0.9977	0.9985	0.9986	0.9957	0.9992	0.9996	0.9996
	\(\hat {s}\)	0.0244	0.0061	0.0025	0.0014	0.0171	0.0033	0.0010	0.0005
α= 0.2	\(\hat {\alpha }\)	0.1925	0.0732	0.0220	0.0037	0.1686	0.0529	0.0113	0.0010
	\(\hat {\rho }\)	0.9902	0.9970	0.9981	0.9982	0.9946	0.9989	0.9994	0.9995
	\(\hat {s}\)	0.0270	0.0069	0.0029	0.0016	0.0189	0.0038	0.0012	0.0006

The bolded values are \({\hat \alpha }\) within \(\pm 2 \sqrt {\alpha (1-\alpha)/M}\) and \({\hat \rho }\) within \(\pm 2 \sqrt {\rho (1-\rho)/M}\)

Back to article page