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 33 Performance of tolerance intervals based on lognormal distribution when the true distribution is gamma (G: Lognormal; 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	\(\hat {\alpha }\)	0.0107	0.0102	0.0110	0.0101	0.0123	0.0124	0.0139	0.0132
	\(\hat {\rho }\)	0.9912	0.9768	0.9626	0.9492	0.9964	0.9906	0.9842	0.9776
	\(\hat {s}\)	0.0206	0.0212	0.0202	0.0172	0.0117	0.0112	0.0110	0.0097
α= 0.05	\(\hat {\alpha }\)	0.0525	0.0516	0.0483	0.0427	0.0555	0.0591	0.0555	0.0578
	\(\hat {\rho }\)	0.9750	0.9598	0.9474	0.9368	0.9883	0.9821	0.9762	0.9707
	\(\hat {s}\)	0.0394	0.0306	0.0251	0.0197	0.0246	0.0178	0.0148	0.0118
α= 0.1	\(\hat {\alpha }\)	0.1008	0.1006	0.0934	0.0878	0.1059	0.1071	0.1106	0.1050
	\(\hat {\rho }\)	0.9610	0.9483	0.9380	0.9296	0.9806	0.9758	0.9709	0.9666
	\(\hat {s}\)	0.0513	0.0358	0.0277	0.0210	0.0338	0.0219	0.0170	0.0130
α= 0.2	\(\hat {\alpha }\)	0.1985	0.1915	0.1843	0.1756	0.2062	0.2051	0.2033	0.2010
	\(\hat {\rho }\)	0.9385	0.9320	0.9256	0.9204	0.9671	0.9663	0.9636	0.9610
	\(\hat {s}\)	0.0663	0.0421	0.0307	0.0225	0.0464	0.0273	0.0198	0.0144
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0145	0.0244	0.0355	0.0491	0.0158	0.0320	0.0517	0.0856
	\(\hat {\rho }\)	0.9993	0.9983	0.9969	0.9954	0.9996	0.9991	0.9983	0.9974
	\(\hat {s}\)	0.0044	0.0031	0.0031	0.0028	0.0031	0.0020	0.0019	0.0017
α= 0.05	\(\hat {\alpha }\)	0.0680	0.0912	0.1153	0.1452	0.0764	0.1126	0.1587	0.2237
	\(\hat {\rho }\)	0.9971	0.9961	0.9948	0.9935	0.9982	0.9978	0.9970	0.9963
	\(\hat {s}\)	0.0103	0.0058	0.0047	0.0037	0.0076	0.0038	0.0030	0.0024
α= 0.1	\(\hat {\alpha }\)	0.1306	0.1579	0.1914	0.2323	0.1452	0.1900	0.2500	0.3291
	\(\hat {\rho }\)	0.9946	0.9943	0.9933	0.9923	0.9966	0.9967	0.9961	0.9955
	\(\hat {s}\)	0.0151	0.0077	0.0057	0.0043	0.0113	0.0052	0.0038	0.0027
α= 0.2	\(\hat {\alpha }\)	0.2376	0.2764	0.3133	0.3668	0.2550	0.3230	0.3834	0.4750
	\(\hat {\rho }\)	0.9898	0.9914	0.9911	0.9907	0.9932	0.9948	0.9947	0.9945
	\(\hat {s}\)	0.0226	0.0104	0.0072	0.0050	0.0174	0.0072	0.0048	0.0033

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