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 4 Performance of tolerance intervals based on Cauchy distribution when the true distribution is Cauchy (F=G: Cauchy)

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	\(\hat {\alpha }\)	0.0980	0.0416	0.0278	0.0161	0.0988	0.0423	0.0291	0.0169
	\(\hat {\rho }\)	0.9405	0.9369	0.9311	0.9251	0.9702	0.9685	0.9656	0.9626
	\(\hat {s}\)	0.0326	0.0188	0.0142	0.0107	0.0166	0.0095	0.0072	0.0054
α= 0.05	\(\hat {\alpha }\)	0.1500	0.0885	0.0650	0.0495	0.1498	0.0901	0.0658	0.0511
	\(\hat {\rho }\)	0.9317	0.9299	0.9250	0.9200	0.9658	0.9649	0.9625	0.9600
	\(\hat {s}\)	0.0372	0.0209	0.0155	0.0114	0.0190	0.0105	0.0078	0.0057
α= 0.1	\(\hat {\alpha }\)	0.1868	0.1226	0.0967	0.0814	0.1871	0.1237	0.0975	0.0833
	\(\hat {\rho }\)	0.9262	0.9256	0.9215	0.9171	0.9630	0.9628	0.9607	0.9586
	\(\hat {s}\)	0.0400	0.0222	0.0162	0.0118	0.0205	0.0112	0.0082	0.0059
α= 0.2	\(\hat {\alpha }\)	0.2423	0.1779	0.1517	0.1381	0.2425	0.1797	0.1529	0.1394
	\(\hat {\rho }\)	0.9187	0.9200	0.9169	0.9135	0.9592	0.9600	0.9585	0.9567
	\(\hat {s}\)	0.0439	0.0238	0.0171	0.0123	0.0225	0.0120	0.0086	0.0062
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0989	0.0423	0.0293	0.0170	0.0989	0.0423	0.0293	0.0170
	\(\hat {\rho }\)	0.9940	0.9937	0.9931	0.9925	0.9970	0.9968	0.9966	0.9963
	\(\hat {s}\)	0.0033	0.0019	0.0014	0.0011	0.0017	0.0010	0.0007	0.0005
α= 0.05	\(\hat {\alpha }\)	0.1502	0.0902	0.0661	0.0515	0.1502	0.0902	0.0661	0.0516
	\(\hat {\rho }\)	0.9932	0.9930	0.9925	0.9920	0.9966	0.9965	0.9963	0.9960
	\(\hat {s}\)	0.0038	0.0021	0.0016	0.0012	0.0019	0.0011	0.0008	0.0006
α= 0.1	\(\hat {\alpha }\)	0.1870	0.1247	0.0977	0.0839	0.1870	0.1248	0.0977	0.0839
	\(\hat {\rho }\)	0.9926	0.9926	0.9921	0.9917	0.9963	0.9963	0.9961	0.9959
	\(\hat {s}\)	0.0041	0.0022	0.0016	0.0012	0.0021	0.0011	0.0008	0.0006
α= 0.2	\(\hat {\alpha }\)	0.2427	0.1800	0.1533	0.1403	0.2426	0.1800	0.1533	0.1403
	\(\hat {\rho }\)	0.9918	0.9920	0.9917	0.9914	0.9959	0.9960	0.9958	0.9957
	\(\hat {s}\)	0.0046	0.0024	0.0017	0.0012	0.0023	0.0012	0.0009	0.0006

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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