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 19 Performance of parametric tolerance intervals based on the model selection approach for skewed distributions when the true distribution is Weibull

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	\(\hat {\alpha }\)	0.0069	0.0089	0.0065	0.0064	0.0074	0.0090	0.0066	0.0069
	\(\hat {\rho }\)	0.9969	0.9971	0.9971	0.9975	0.9979	0.9981	0.9981	0.9984
	\(\hat {s}\)	0.0325	0.0294	0.0325	0.0263	0.0280	0.0252	0.0293	0.0221
α= 0.05	\(\hat {\alpha }\)	0.0359	0.0364	0.0348	0.0322	0.0361	0.0372	0.0357	0.0330
	\(\hat {\rho }\)	0.9870	0.9869	0.9878	0.9883	0.9916	0.9916	0.9922	0.9927
	\(\hat {s}\)	0.0583	0.0573	0.0562	0.0526	0.0483	0.0466	0.0467	0.0426
α= 0.1	\(\hat {\alpha }\)	0.0726	0.0703	0.0646	0.0657	0.0746	0.0721	0.0655	0.0668
	\(\hat {\rho }\)	0.9754	0.9755	0.9767	0.9770	0.9840	0.9840	0.9850	0.9853
	\(\hat {s}\)	0.0781	0.0780	0.0756	0.0729	0.0642	0.0637	0.0621	0.0589
α= 0.2	\(\hat {\alpha }\)	0.1364	0.1331	0.1334	0.1324	0.1390	0.1353	0.1361	0.1351
	\(\hat {\rho }\)	0.9537	0.9541	0.9556	0.9557	0.9692	0.9695	0.9708	0.9709
	\(\hat {s}\)	0.1053	0.1051	0.1020	0.1006	0.0872	0.0872	0.0844	0.0823
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0075	0.0089	0.0070	0.0071	0.0072	0.0088	0.0069	0.0071
	\(\hat {\rho }\)	0.9989	0.9990	0.9989	0.9992	0.9991	0.9992	0.9990	0.9994
	\(\hat {s}\)	0.0214	0.0208	0.0255	0.0165	0.0190	0.0195	0.0243	0.0148
α= 0.05	\(\hat {\alpha }\)	0.0370	0.0371	0.0356	0.0338	0.0373	0.0370	0.0354	0.0338
	\(\hat {\rho }\)	0.9958	0.9959	0.9961	0.9965	0.9966	0.9967	0.9968	0.9972
	\(\hat {s}\)	0.0366	0.0342	0.0361	0.0311	0.0336	0.0313	0.0338	0.0283
α= 0.1	\(\hat {\alpha }\)	0.0742	0.0721	0.0666	0.0681	0.0737	0.0720	0.0660	0.0680
	\(\hat {\rho }\)	0.9920	0.9920	0.9925	0.9930	0.9935	0.9936	0.9940	0.9944
	\(\hat {s}\)	0.0475	0.0459	0.0461	0.0421	0.0434	0.0416	0.0424	0.0381
α= 0.2	\(\hat {\alpha }\)	0.1397	0.1369	0.1363	0.1357	0.1397	0.1367	0.1364	0.1359
	\(\hat {\rho }\)	0.9843	0.9844	0.9854	0.9856	0.9873	0.9874	0.9882	0.9885
	\(\hat {s}\)	0.0638	0.0634	0.0619	0.0588	0.0578	0.0572	0.0563	0.0529

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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