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 30 Performance of tolerance intervals based on Laplace distribution when the true distribution is logistic (G: Laplace; 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	\(\hat {\alpha }\)	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	\(\hat {\rho }\)	0.9999	0.9946	0.9846	0.9722	1.0000	0.9988	0.9960	0.9917
	\(\hat {s}\)	0.0010	0.0065	0.0093	0.0101	0.0003	0.0020	0.0032	0.0040
α= 0.05	\(\hat {\alpha }\)	0.0036	0.0019	0.0009	0.0006	0.0026	0.0011	0.0002	0.0001
	\(\hat {\rho }\)	0.9950	0.9833	0.9726	0.9614	0.9986	0.9953	0.9917	0.9874
	\(\hat {s}\)	0.0139	0.0149	0.0142	0.0129	0.0062	0.0058	0.0057	0.0055
α= 0.1	\(\hat {\alpha }\)	0.0182	0.0091	0.0042	0.0018	0.0142	0.0049	0.0015	0.0006
	\(\hat {\rho }\)	0.9863	0.9745	0.9650	0.9551	0.9955	0.9921	0.9887	0.9848
	\(\hat {s}\)	0.0263	0.0203	0.0170	0.0143	0.0130	0.0086	0.0073	0.0064
α= 0.2	\(\hat {\alpha }\)	0.0612	0.0338	0.0174	0.0079	0.0490	0.0211	0.0071	0.0019
	\(\hat {\rho }\)	0.9693	0.9616	0.9549	0.9473	0.9883	0.9868	0.9844	0.9813
	\(\hat {s}\)	0.0431	0.0269	0.0202	0.0159	0.0238	0.0126	0.0094	0.0075
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
	\(\hat {\rho }\)	1.0000	1.0000	0.9998	0.9995	1.0000	1.0000	0.9999	0.9999
	\(\hat {s}\)	0.0000	0.0001	0.0003	0.0004	0.0000	0.0001	0.0001	0.0001
α= 0.05	\(\hat {\alpha }\)	0.0021	0.0005	0.0000	0.0000	0.0018	0.0003	0.0000	0.0000
	\(\hat {\rho }\)	0.9999	0.9997	0.9995	0.9991	1.0000	0.9999	0.9998	0.9997
	\(\hat {s}\)	0.0012	0.0007	0.0006	0.0007	0.0006	0.0003	0.0002	0.0003
α= 0.1	\(\hat {\alpha }\)	0.0101	0.0021	0.0003	0.0000	0.0089	0.0014	0.0001	0.0000
	\(\hat {\rho }\)	0.9995	0.9994	0.9992	0.9988	0.9998	0.9998	0.9997	0.9996
	\(\hat {s}\)	0.0031	0.0012	0.0009	0.0008	0.0018	0.0005	0.0004	0.0003
α= 0.2	\(\hat {\alpha }\)	0.0353	0.0093	0.0018	0.0003	0.0312	0.0072	0.0009	0.0001
	\(\hat {\rho }\)	0.9984	0.9988	0.9987	0.9983	0.9992	0.9995	0.9995	0.9994
	\(\hat {s}\)	0.0067	0.0020	0.0013	0.0011	0.0041	0.0010	0.0006	0.0004

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