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 35 Performance of tolerance intervals based on gamma distribution when the true distribution is Weibull (G: Gamma; 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	\(\hat {\alpha }\)	0.0184	0.0190	0.0134	0.0084	0.0243	0.0314	0.0274	0.0219
	\(\hat {\rho }\)	0.9862	0.9717	0.9609	0.9507	0.9934	0.9862	0.9807	0.9753
	\(\hat {s}\)	0.0267	0.0243	0.0207	0.0170	0.0171	0.0146	0.0124	0.0103
α= 0.05	\(\hat {\alpha }\)	0.0728	0.0662	0.0530	0.0354	0.0870	0.0930	0.0877	0.0701
	\(\hat {\rho }\)	0.9671	0.9553	0.9471	0.9397	0.9826	0.9770	0.9731	0.9693
	\(\hat {s}\)	0.0454	0.0332	0.0255	0.0195	0.0308	0.0210	0.0158	0.0121
α= 0.1	\(\hat {\alpha }\)	0.1338	0.1209	0.0986	0.0677	0.1574	0.1588	0.1411	0.1164
	\(\hat {\rho }\)	0.9519	0.9442	0.9386	0.9332	0.9734	0.9707	0.9683	0.9657
	\(\hat {s}\)	0.0571	0.0382	0.0281	0.0208	0.0399	0.0249	0.0178	0.0132
α= 0.2	\(\hat {\alpha }\)	0.2529	0.2176	0.1778	0.1332	0.2769	0.2610	0.2310	0.2028
	\(\hat {\rho }\)	0.9283	0.9287	0.9272	0.9249	0.9585	0.9615	0.9617	0.9609
	\(\hat {s}\)	0.0717	0.0444	0.0313	0.0224	0.0523	0.0300	0.0204	0.0145
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0410	0.0800	0.1212	0.1808	0.0462	0.1072	0.1755	0.3005
	\(\hat {\rho }\)	0.9983	0.9965	0.9949	0.9931	0.9989	0.9980	0.9970	0.9958
	\(\hat {s}\)	0.0081	0.0055	0.0047	0.0040	0.0064	0.0038	0.0032	0.0027
α= 0.05	\(\hat {\alpha }\)	0.1381	0.2099	0.2633	0.3537	0.1588	0.2609	0.3565	0.5027
	\(\hat {\rho }\)	0.9944	0.9932	0.9921	0.9908	0.9963	0.9957	0.9951	0.9942
	\(\hat {s}\)	0.0155	0.0089	0.0064	0.0049	0.0122	0.0064	0.0045	0.0035
α= 0.1	\(\hat {\alpha }\)	0.2319	0.3033	0.3665	0.4577	0.2608	0.3714	0.4733	0.6154
	\(\hat {\rho }\)	0.9908	0.9908	0.9903	0.9895	0.9937	0.9940	0.9938	0.9933
	\(\hat {s}\)	0.0209	0.0110	0.0075	0.0054	0.0167	0.0081	0.0054	0.0039
α= 0.2	\(\hat {\alpha }\)	0.3680	0.4369	0.5039	0.5919	0.4112	0.5082	0.6106	0.7335
	\(\hat {\rho }\)	0.9844	0.9872	0.9878	0.9877	0.9889	0.9914	0.9920	0.9919
	\(\hat {s}\)	0.0288	0.0139	0.0088	0.0061	0.0234	0.0105	0.0065	0.0045

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