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 7 Performance of tolerance intervals based on logistic distribution when the true distribution is logistic (F=G: 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	\(\hat {\alpha }\)	0.0138	0.0127	0.0135	0.0114	0.0133	0.0123	0.0129	0.0111
	\(\hat {\rho }\)	0.9908	0.9765	0.9615	0.9472	0.9966	0.9919	0.9857	0.9787
	\(\hat {s}\)	0.0231	0.0223	0.0205	0.0172	0.0128	0.0111	0.0108	0.0096
α= 0.05	\(\hat {\alpha }\)	0.0675	0.0638	0.0582	0.0593	0.0658	0.0620	0.0570	0.0566
	\(\hat {\rho }\)	0.9715	0.9569	0.9444	0.9336	0.9875	0.9824	0.9767	0.9710
	\(\hat {s}\)	0.0444	0.0322	0.0255	0.0196	0.0277	0.0185	0.0149	0.0118
α= 0.1	\(\hat {\alpha }\)	0.1290	0.1206	0.1158	0.1136	0.1264	0.1184	0.1132	0.1091
	\(\hat {\rho }\)	0.9545	0.9438	0.9341	0.9259	0.9782	0.9752	0.9708	0.9664
	\(\hat {s}\)	0.0573	0.0375	0.0280	0.0209	0.0381	0.0230	0.0173	0.0130
α= 0.2	\(\hat {\alpha }\)	0.2439	0.2277	0.2210	0.2210	0.2408	0.2223	0.2162	0.2142
	\(\hat {\rho }\)	0.9276	0.9256	0.9206	0.9160	0.9619	0.9644	0.9625	0.9602
	\(\hat {s}\)	0.0726	0.0436	0.0310	0.0223	0.0521	0.0288	0.0202	0.0144
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0127	0.0114	0.0118	0.0089	0.0122	0.0113	0.0110	0.0084
	\(\hat {\rho }\)	0.9994	0.9992	0.9985	0.9974	0.9997	0.9996	0.9994	0.9990
	\(\hat {s}\)	0.0043	0.0023	0.0023	0.0020	0.0030	0.0013	0.0012	0.0010
α= 0.05	\(\hat {\alpha }\)	0.0622	0.0581	0.0523	0.0497	0.0608	0.0565	0.0506	0.0469
	\(\hat {\rho }\)	0.9973	0.9974	0.9967	0.9957	0.9985	0.9988	0.9985	0.9981
	\(\hat {s}\)	0.0111	0.0051	0.0039	0.0030	0.0080	0.0030	0.0022	0.0016
α= 0.1	\(\hat {\alpha }\)	0.1217	0.1128	0.1060	0.1009	0.1206	0.1109	0.1028	0.0964
	\(\hat {\rho }\)	0.9947	0.9958	0.9953	0.9946	0.9968	0.9979	0.9978	0.9975
	\(\hat {s}\)	0.0167	0.0072	0.0050	0.0036	0.0124	0.0044	0.0029	0.0020
α= 0.2	\(\hat {\alpha }\)	0.2341	0.2130	0.2072	0.1990	0.2309	0.2101	0.2028	0.1932
	\(\hat {\rho }\)	0.9892	0.9928	0.9931	0.9929	0.9932	0.9962	0.9966	0.9966
	\(\hat {s}\)	0.0255	0.0103	0.0065	0.0043	0.0194	0.0066	0.0040	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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