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 36 Performance of tolerance intervals based on lognormal distribution when the true distribution is Weibull (G: Lognormal; F: 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	\(\hat {\alpha }\)	0.0195	0.0215	0.0134	0.0063	0.0344	0.0545	0.0565	0.0503
	\(\hat {\rho }\)	0.9826	0.9678	0.9592	0.9520	0.9902	0.9810	0.9754	0.9707
	\(\hat {s}\)	0.0269	0.0248	0.0207	0.0168	0.0183	0.0168	0.0140	0.0113
α= 0.05	\(\hat {\alpha }\)	0.0700	0.0633	0.0455	0.0229	0.1066	0.1316	0.1246	0.1128
	\(\hat {\rho }\)	0.9657	0.9547	0.9490	0.9441	0.9792	0.9723	0.9688	0.9658
	\(\hat {s}\)	0.0427	0.0322	0.0248	0.0191	0.0304	0.0221	0.0167	0.0127
α= 0.1	\(\hat {\alpha }\)	0.1200	0.1056	0.0766	0.0422	0.1743	0.1945	0.1781	0.1594
	\(\hat {\rho }\)	0.9536	0.9464	0.9428	0.9395	0.9710	0.9669	0.9649	0.9630
	\(\hat {s}\)	0.0525	0.0366	0.0272	0.0204	0.0381	0.0253	0.0183	0.0135
α= 0.2	\(\hat {\alpha }\)	0.2089	0.1773	0.1310	0.0753	0.2744	0.2818	0.2587	0.2329
	\(\hat {\rho }\)	0.9353	0.9347	0.9344	0.9334	0.9587	0.9595	0.9598	0.9594
	\(\hat {s}\)	0.0654	0.0424	0.0303	0.0220	0.0484	0.0295	0.0204	0.0146
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0842	0.2270	0.3715	0.5708	0.1152	0.3247	0.5461	0.7836
	\(\hat {\rho }\)	0.9965	0.9929	0.9904	0.9881	0.9976	0.9950	0.9932	0.9915
	\(\hat {s}\)	0.0094	0.0081	0.0069	0.0057	0.0075	0.0062	0.0052	0.0044
α= 0.05	\(\hat {\alpha }\)	0.2350	0.4073	0.5601	0.7285	0.2983	0.5262	0.7235	0.8901
	\(\hat {\rho }\)	0.9914	0.9886	0.9870	0.9855	0.9937	0.9917	0.9906	0.9895
	\(\hat {s}\)	0.0167	0.0115	0.0086	0.0066	0.0136	0.0090	0.0068	0.0052
α= 0.1	\(\hat {\alpha }\)	0.3410	0.5074	0.6525	0.7959	0.4173	0.6331	0.7938	0.9264
	\(\hat {\rho }\)	0.9873	0.9858	0.9849	0.9840	0.9905	0.9895	0.9890	0.9883
	\(\hat {s}\)	0.0217	0.0135	0.0097	0.0071	0.0179	0.0108	0.0076	0.0056
α= 0.2	\(\hat {\alpha }\)	0.4770	0.6272	0.7471	0.8629	0.5547	0.7397	0.8622	0.9588
	\(\hat {\rho }\)	0.9809	0.9819	0.9823	0.9821	0.9854	0.9864	0.9868	0.9868
	\(\hat {s}\)	0.0287	0.0162	0.0109	0.0078	0.0239	0.0131	0.0087	0.0062

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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