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 38 Performance of tolerance intervals based on Weibull distribution when the true distribution is lognormal (G: Weibull; F: Lognormal)

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	\(\hat {\alpha }\)	0.0114	0.0064	0.0021	0.0004	0.0191	0.0149	0.0102	0.0047
	\(\hat {\rho }\)	0.9915	0.9846	0.9777	0.9705	0.9951	0.9916	0.9877	0.9835
	\(\hat {s}\)	0.0226	0.0181	0.0158	0.0135	0.0163	0.0120	0.0104	0.0089
α= 0.05	\(\hat {\alpha }\)	0.0433	0.0196	0.0086	0.0019	0.0648	0.0460	0.0308	0.0153
	\(\hat {\rho }\)	0.9796	0.9743	0.9688	0.9631	0.9874	0.9851	0.9821	0.9789
	\(\hat {s}\)	0.0376	0.0252	0.0198	0.0157	0.0281	0.0175	0.0134	0.0105
α= 0.1	\(\hat {\alpha }\)	0.0722	0.0360	0.0161	0.0037	0.1062	0.0777	0.0516	0.0276
	\(\hat {\rho }\)	0.9703	0.9676	0.9634	0.9589	0.9813	0.9809	0.9788	0.9763
	\(\hat {s}\)	0.0466	0.0292	0.0220	0.0168	0.0354	0.0206	0.0151	0.0114
α= 0.2	\(\hat {\alpha }\)	0.1239	0.0655	0.0304	0.0080	0.1705	0.1273	0.0857	0.0523
	\(\hat {\rho }\)	0.9566	0.9584	0.9565	0.9536	0.9720	0.9751	0.9744	0.9730
	\(\hat {s}\)	0.0578	0.0341	0.0246	0.0182	0.0446	0.0244	0.0171	0.0125
		ρ= 0.99				ρ= 0.995
α = 0.01	\(\hat {\alpha }\)	0.0450	0.0698	0.0923	0.1367	0.0566	0.1061	0.1650	0.2684
	\(\hat {\rho }\)	0.9980	0.9971	0.9958	0.9942	0.9985	0.9980	0.9971	0.9960
	\(\hat {s}\)	0.0096	0.0059	0.0049	0.0042	0.0081	0.0046	0.0037	0.0032
α= 0.05	\(\hat {\alpha }\)	0.1302	0.1664	0.2084	0.2677	0.1566	0.2321	0.3193	0.4482
	\(\hat {\rho }\)	0.9944	0.9942	0.9933	0.9921	0.9957	0.9959	0.9953	0.9944
	\(\hat {s}\)	0.0176	0.0093	0.0068	0.0053	0.0151	0.0074	0.0053	0.0041
α= 0.1	\(\hat {\alpha }\)	0.1958	0.2415	0.2863	0.3544	0.2357	0.3185	0.4176	0.5446
	\(\hat {\rho }\)	0.9912	0.9922	0.9917	0.9908	0.9931	0.9943	0.9941	0.9935
	\(\hat {s}\)	0.0227	0.0113	0.0079	0.0059	0.0197	0.0092	0.0062	0.0046
α= 0.2	\(\hat {\alpha }\)	0.2993	0.3383	0.3897	0.4597	0.3499	0.4338	0.5293	0.6562
	\(\hat {\rho }\)	0.9861	0.9893	0.9895	0.9892	0.9890	0.9920	0.9924	0.9922
	\(\hat {s}\)	0.0294	0.0140	0.0093	0.0066	0.0257	0.0115	0.0074	0.0052

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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