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 32 Performance of tolerance intervals based on normal distribution when the true distribution is logistic (G: Normal; F: Logistic)

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	\(\hat {\alpha }\)	0.0158	0.0193	0.0198	0.0211	0.0207	0.0327	0.0409	0.0540
	\(\hat {\rho }\)	0.9866	0.9685	0.9543	0.9422	0.9937	0.9848	0.9770	0.9698
	\(\hat {s}\)	0.0246	0.0242	0.0213	0.0174	0.0152	0.0145	0.0131	0.0111
α= 0.05	\(\hat {\alpha }\)	0.0666	0.0738	0.0739	0.0675	0.0835	0.1089	0.1330	0.1555
	\(\hat {\rho }\)	0.9674	0.9508	0.9398	0.9311	0.9829	0.9744	0.9680	0.9627
	\(\hat {s}\)	0.0429	0.0325	0.0256	0.0196	0.0287	0.0210	0.0166	0.0129
α= 0.1	\(\hat {\alpha }\)	0.1264	0.1326	0.1317	0.1193	0.1491	0.1875	0.2156	0.2434
	\(\hat {\rho }\)	0.9523	0.9396	0.9313	0.9248	0.9737	0.9673	0.9625	0.9585
	\(\hat {s}\)	0.0538	0.0371	0.0279	0.0207	0.0376	0.0248	0.0186	0.0139
α= 0.2	\(\hat {\alpha }\)	0.2314	0.2370	0.2321	0.2104	0.2668	0.3080	0.3425	0.3651
	\(\hat {\rho }\)	0.9296	0.9245	0.9203	0.9168	0.9587	0.9574	0.9552	0.9532
	\(\hat {s}\)	0.0673	0.0426	0.0306	0.0220	0.0494	0.0296	0.0210	0.0151
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0340	0.0793	0.1558	0.2830	0.0401	0.1124	0.2363	0.4377
	\(\hat {\rho }\)	0.9984	0.9962	0.9939	0.9916	0.9990	0.9977	0.9963	0.9948
	\(\hat {s}\)	0.0065	0.0052	0.0047	0.0042	0.0049	0.0035	0.0032	0.0028
α= 0.05	\(\hat {\alpha }\)	0.1300	0.2299	0.3494	0.5153	0.1526	0.2936	0.4647	0.6784
	\(\hat {\rho }\)	0.9948	0.9927	0.9907	0.9889	0.9965	0.9954	0.9941	0.9929
	\(\hat {s}\)	0.0136	0.0085	0.0066	0.0052	0.0105	0.0061	0.0046	0.0036
α= 0.1	\(\hat {\alpha }\)	0.2237	0.3421	0.4778	0.6390	0.2565	0.4214	0.5887	0.7872
	\(\hat {\rho }\)	0.9912	0.9900	0.9886	0.9872	0.9940	0.9935	0.9927	0.9918
	\(\hat {s}\)	0.0189	0.0107	0.0077	0.0058	0.0148	0.0078	0.0054	0.0041
α= 0.2	\(\hat {\alpha }\)	0.3693	0.4961	0.6246	0.7741	0.4162	0.5753	0.7320	0.8811
	\(\hat {\rho }\)	0.9847	0.9859	0.9855	0.9850	0.9893	0.9906	0.9905	0.9902
	\(\hat {s}\)	0.0267	0.0137	0.0092	0.0065	0.0213	0.0102	0.0066	0.0047

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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