Table 31 Performance of tolerance intervals based on Cauchy distribution when the true distribution is logistic (G: Cauchy; 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 | \(\hat {\alpha }\) | 0.0038 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9978 | 0.9996 | 0.9996 | 0.9995 | 0.9999 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0147 | 0.0014 | 0.0006 | 0.0005 | 0.0059 | 0.0000 | 0.0000 | 0.0000 | |
α= 0.05 | \(\hat {\alpha }\) | 0.0069 | 0.0000 | 0.0000 | 0.0000 | 0.0006 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9963 | 0.9992 | 0.9993 | 0.9991 | 0.9998 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0193 | 0.0023 | 0.0010 | 0.0008 | 0.0066 | 0.0000 | 0.0000 | 0.0000 | |
α= 0.1 | \(\hat {\alpha }\) | 0.0093 | 0.0000 | 0.0000 | 0.0000 | 0.0013 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9949 | 0.9988 | 0.9990 | 0.9989 | 0.9997 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0225 | 0.0030 | 0.0014 | 0.0010 | 0.0072 | 0.0001 | 0.0000 | 0.0000 | |
α= 0.2 | \(\hat {\alpha }\) | 0.0148 | 0.0000 | 0.0000 | 0.0000 | 0.0018 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9927 | 0.9981 | 0.9986 | 0.9984 | 0.9995 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0274 | 0.0042 | 0.0019 | 0.0013 | 0.0080 | 0.0001 | 0.0000 | 0.0000 | |
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 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
α = 0.05 | \(\hat {\alpha }\) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
α = 0.1 | \(\hat {\alpha }\) | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
α = 0.2 | \(\hat {\alpha }\) | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |