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 29 Performance of tolerance intervals based on normal distribution when the true distribution is Laplace (G: Normal; 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	\(\hat {\alpha }\)	0.0365	0.0451	0.0506	0.0498	0.0554	0.0977	0.1347	0.1996
	\(\hat {\rho }\)	0.9775	0.9583	0.9453	0.9347	0.9874	0.9763	0.9681	0.9610
	\(\hat {s}\)	0.0331	0.0295	0.0246	0.0196	0.0229	0.0202	0.0171	0.0140
α= 0.05	\(\hat {\alpha }\)	0.1144	0.1290	0.1276	0.1233	0.1635	0.2249	0.2842	0.3612
	\(\hat {\rho }\)	0.9554	0.9409	0.9320	0.9248	0.9729	0.9645	0.9587	0.9539
	\(\hat {s}\)	0.0504	0.0367	0.0282	0.0214	0.0372	0.0265	0.0204	0.0156
α= 0.1	\(\hat {\alpha }\)	0.1866	0.1956	0.1917	0.1840	0.2503	0.3154	0.3837	0.4557
	\(\hat {\rho }\)	0.9400	0.9305	0.9244	0.9193	0.9622	0.9571	0.9532	0.9499
	\(\hat {s}\)	0.0599	0.0405	0.0301	0.0223	0.0456	0.0300	0.0222	0.0165
α= 0.2	\(\hat {\alpha }\)	0.2977	0.2958	0.2916	0.2769	0.3787	0.4428	0.5053	0.5743
	\(\hat {\rho }\)	0.9184	0.9170	0.9148	0.9125	0.9465	0.9472	0.9462	0.9448
	\(\hat {s}\)	0.0712	0.0450	0.0323	0.0234	0.0561	0.0343	0.0243	0.0176
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.1117	0.2661	0.4679	0.7163	0.1408	0.3503	0.6016	0.8563
	\(\hat {\rho }\)	0.9956	0.9918	0.9886	0.9856	0.9969	0.9944	0.9922	0.9901
	\(\hat {s}\)	0.0118	0.0095	0.0081	0.0068	0.0094	0.0072	0.0061	0.0051
α= 0.05	\(\hat {\alpha }\)	0.2813	0.4762	0.6635	0.8562	0.3308	0.5766	0.7783	0.9413
	\(\hat {\rho }\)	0.9890	0.9864	0.9842	0.9821	0.9920	0.9904	0.9889	0.9874
	\(\hat {s}\)	0.0210	0.0137	0.0104	0.0080	0.0172	0.0107	0.0080	0.0062
α= 0.1	\(\hat {\alpha }\)	0.3956	0.5923	0.7564	0.9062	0.4552	0.6824	0.8500	0.9658
	\(\hat {\rho }\)	0.9837	0.9827	0.9814	0.9801	0.9878	0.9876	0.9868	0.9859
	\(\hat {s}\)	0.0271	0.0162	0.0117	0.0087	0.0225	0.0128	0.0091	0.0067
α= 0.2	\(\hat {\alpha }\)	0.5430	0.7129	0.8442	0.9480	0.6036	0.7896	0.9134	0.9828
	\(\hat {\rho }\)	0.9752	0.9776	0.9778	0.9774	0.9811	0.9836	0.9840	0.9838
	\(\hat {s}\)	0.0352	0.0194	0.0133	0.0095	0.0297	0.0156	0.0105	0.0074

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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