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 34 Performance of tolerance intervals based on Weibull distribution when the true distribution is gamma (G: Weibull; F: Gamma)

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	\(\hat {\alpha }\)	0.0074	0.0040	0.0020	0.0003	0.0106	0.0084	0.0055	0.0015
	\(\hat {\rho }\)	0.9948	0.9875	0.9784	0.9681	0.9975	0.9947	0.9907	0.9857
	\(\hat {s}\)	0.0177	0.0162	0.0154	0.0137	0.0116	0.0094	0.0088	0.0080
α= 0.05	\(\hat {\alpha }\)	0.0307	0.0185	0.0090	0.0026	0.0384	0.0276	0.0195	0.0110
	\(\hat {\rho }\)	0.9844	0.9761	0.9674	0.9584	0.9919	0.9888	0.9849	0.9804
	\(\hat {s}\)	0.0330	0.0242	0.0200	0.0162	0.0227	0.0150	0.0122	0.0099
α= 0.1	\(\hat {\alpha }\)	0.0570	0.0334	0.0180	0.0082	0.0714	0.0509	0.0379	0.0245
	\(\hat {\rho }\)	0.9753	0.9682	0.9606	0.9529	0.9866	0.9845	0.9812	0.9773
	\(\hat {s}\)	0.0426	0.0288	0.0224	0.0174	0.0300	0.0184	0.0141	0.0110
α= 0.2	\(\hat {\alpha }\)	0.1054	0.0661	0.0412	0.0208	0.1283	0.0986	0.0704	0.0491
	\(\hat {\rho }\)	0.9608	0.9571	0.9518	0.9460	0.9779	0.9783	0.9761	0.9733
	\(\hat {s}\)	0.0547	0.0342	0.0253	0.0189	0.0397	0.0227	0.0164	0.0122
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0183	0.0231	0.0280	0.0359	0.0218	0.0318	0.0439	0.0707
	\(\hat {\rho }\)	0.9992	0.9988	0.9979	0.9966	0.9994	0.9993	0.9988	0.9980
	\(\hat {s}\)	0.0058	0.0036	0.0031	0.0028	0.0047	0.0026	0.0021	0.0019
α= 0.05	\(\hat {\alpha }\)	0.0646	0.0743	0.0812	0.0992	0.0752	0.0995	0.1259	0.1794
	\(\hat {\rho }\)	0.9971	0.9970	0.9961	0.9949	0.9979	0.9981	0.9976	0.9968
	\(\hat {s}\)	0.0123	0.0064	0.0048	0.0038	0.0101	0.0048	0.0034	0.0027
α= 0.1	\(\hat {\alpha }\)	0.1111	0.1236	0.1348	0.1551	0.1281	0.1615	0.1972	0.2591
	\(\hat {\rho }\)	0.9950	0.9955	0.9949	0.9938	0.9964	0.9970	0.9967	0.9961
	\(\hat {s}\)	0.0169	0.0083	0.0058	0.0044	0.0140	0.0064	0.0043	0.0031
α= 0.2	\(\hat {\alpha }\)	0.1916	0.2031	0.2188	0.2484	0.2170	0.2515	0.3006	0.3778
	\(\hat {\rho }\)	0.9913	0.9932	0.9930	0.9923	0.9935	0.9954	0.9955	0.9951
	\(\hat {s}\)	0.0232	0.0109	0.0072	0.0051	0.0195	0.0085	0.0054	0.0037

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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