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 13 Performance of tolerance intervals based on logistic distribution when the true distribution is Laplace (G: Logistic; F: Laplace)

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	\(\hat {\alpha }\)	0.1700	0.1078	0.0797	0.0899	0.2058	0.1592	0.1499	0.1875
	\(\hat {\rho }\)	0.9425	0.9542	0.9459	0.9338	0.9587	0.9735	0.9696	0.9627
	\(\hat {s}\)	0.1509	0.0424	0.0303	0.0232	0.1464	0.0292	0.0203	0.0157
α= 0.05	\(\hat {\alpha }\)	0.2321	0.1795	0.1531	0.1670	0.2771	0.2454	0.2546	0.3142
	\(\hat {\rho }\)	0.9341	0.9390	0.9326	0.9229	0.9565	0.9639	0.9614	0.9559
	\(\hat {s}\)	0.0841	0.0471	0.0327	0.0243	0.0648	0.0333	0.0225	0.0168
α= 0.1	\(\hat {\alpha }\)	0.2783	0.2265	0.2080	0.2273	0.3307	0.3055	0.3277	0.3944
	\(\hat {\rho }\)	0.9231	0.9299	0.9251	0.9170	0.9492	0.9581	0.9566	0.9521
	\(\hat {s}\)	0.0882	0.0493	0.0337	0.0247	0.0684	0.0352	0.0236	0.0173
α= 0.2	\(\hat {\alpha }\)	0.3446	0.2995	0.2926	0.3174	0.4005	0.3888	0.4205	0.5022
	\(\hat {\rho }\)	0.9084	0.9182	0.9157	0.9097	0.9392	0.9506	0.9506	0.9474
	\(\hat {s}\)	0.0925	0.0515	0.0348	0.0252	0.0724	0.0374	0.0247	0.0179
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.2565	0.2426	0.2912	0.4084	0.2703	0.2696	0.3324	0.4795
	\(\hat {\rho }\)	0.9773	0.9920	0.9917	0.9899	0.9812	0.9951	0.9952	0.9942
	\(\hat {s}\)	0.1397	0.0125	0.0078	0.0060	0.1386	0.0087	0.0051	0.0039
α= 0.05	\(\hat {\alpha }\)	0.3451	0.3594	0.4284	0.5760	0.3622	0.3911	0.4805	0.6445
	\(\hat {\rho }\)	0.9819	0.9886	0.9889	0.9875	0.9872	0.9929	0.9934	0.9927
	\(\hat {s}\)	0.0369	0.0148	0.0091	0.0067	0.0294	0.0105	0.0061	0.0044
α= 0.1	\(\hat {\alpha }\)	0.3995	0.4290	0.5143	0.6621	0.4199	0.4612	0.5610	0.7264
	\(\hat {\rho }\)	0.9788	0.9865	0.9872	0.9862	0.9850	0.9916	0.9924	0.9918
	\(\hat {s}\)	0.0392	0.0161	0.0098	0.0071	0.0314	0.0115	0.0066	0.0047
α= 0.2	\(\hat {\alpha }\)	0.4771	0.5192	0.6102	0.7576	0.4973	0.5524	0.6583	0.8099
	\(\hat {\rho }\)	0.9745	0.9837	0.9851	0.9845	0.9820	0.9897	0.9910	0.9908
	\(\hat {s}\)	0.0420	0.0175	0.0106	0.0075	0.0337	0.0126	0.0073	0.0051

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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