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 27 Performance of tolerance intervals based on Laplace distribution when the true distribution is Cauchy (G: Laplace; F: 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	\(\hat {\alpha }\)	0.0048	0.0579	0.0642	0.0389	0.0433	0.2811	0.3342	0.3407
	\(\hat {\rho }\)	0.9738	0.9506	0.9462	0.9462	0.9797	0.9616	0.9583	0.9583
	\(\hat {s}\)	0.0188	0.0292	0.0284	0.0257	0.0145	0.0227	0.0221	0.0199
α= 0.05	\(\hat {\alpha }\)	0.1142	0.1514	0.1207	0.0694	0.3095	0.4305	0.4381	0.4093
	\(\hat {\rho }\)	0.9467	0.9381	0.9388	0.9416	0.9587	0.9519	0.9525	0.9548
	\(\hat {s}\)	0.0379	0.0365	0.0323	0.0278	0.0295	0.0284	0.0251	0.0216
α= 0.1	\(\hat {\alpha }\)	0.2058	0.2058	0.1542	0.0882	0.4380	0.4940	0.4796	0.4430
	\(\hat {\rho }\)	0.9330	0.9316	0.9349	0.9392	0.9479	0.9469	0.9495	0.9530
	\(\hat {s}\)	0.0474	0.0402	0.0343	0.0290	0.0370	0.0313	0.0267	0.0225
α= 0.2	\(\hat {\alpha }\)	0.3135	0.2739	0.1998	0.1133	0.5529	0.5540	0.5264	0.4814
	\(\hat {\rho }\)	0.9171	0.9241	0.9304	0.9365	0.9355	0.9411	0.9461	0.9509
	\(\hat {s}\)	0.0582	0.0445	0.0366	0.0303	0.0456	0.0347	0.0285	0.0235
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.5641	0.8403	0.8835	0.9014	0.7757	0.9216	0.9420	0.9547
	\(\hat {\rho }\)	0.9867	0.9748	0.9726	0.9727	0.9885	0.9781	0.9762	0.9762
	\(\hat {s}\)	0.0095	0.0149	0.0145	0.0131	0.0083	0.0130	0.0127	0.0114
α= 0.05	\(\hat {\alpha }\)	0.8194	0.8798	0.9032	0.9125	0.9105	0.9412	0.9516	0.9591
	\(\hat {\rho }\)	0.9729	0.9684	0.9689	0.9704	0.9764	0.9725	0.9729	0.9742
	\(\hat {s}\)	0.0194	0.0187	0.0165	0.0142	0.0169	0.0163	0.0144	0.0123
α= 0.1	\(\hat {\alpha }\)	0.8676	0.8933	0.9108	0.9179	0.9314	0.9467	0.9546	0.9605
	\(\hat {\rho }\)	0.9658	0.9651	0.9669	0.9692	0.9702	0.9697	0.9712	0.9732
	\(\hat {s}\)	0.0245	0.0206	0.0175	0.0147	0.0213	0.0180	0.0153	0.0128
α= 0.2	\(\hat {\alpha }\)	0.8996	0.9082	0.9182	0.9220	0.9466	0.9538	0.9582	0.9620
	\(\hat {\rho }\)	0.9576	0.9614	0.9647	0.9679	0.9631	0.9664	0.9693	0.9721
	\(\hat {s}\)	0.0302	0.0229	0.0187	0.0154	0.0264	0.0199	0.0163	0.0134

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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