Table 28 Performance of tolerance intervals based on Cauchy distribution when the true distribution is Laplace (G: Cauchy; 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 | \(\hat {\alpha }\) | 0.0092 | 0.0001 | 0.0000 | 0.0000 | 0.0007 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9940 | 0.9967 | 0.9965 | 0.9952 | 0.9996 | 1.0000 | 1.0000 | 1.0000 | |
\(\hat {s}\) | 0.0193 | 0.0062 | 0.0040 | 0.0034 | 0.0035 | 0.0003 | 0.0001 | 0.0001 | |
α= 0.05 | \(\hat {\alpha }\) | 0.0178 | 0.0005 | 0.0000 | 0.0000 | 0.0026 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9905 | 0.9947 | 0.9946 | 0.9933 | 0.9993 | 0.9999 | 0.9999 | 0.9999 | |
\(\hat {s}\) | 0.0256 | 0.0088 | 0.0055 | 0.0044 | 0.0050 | 0.0005 | 0.0002 | 0.0001 | |
α= 0.1 | \(\hat {\alpha }\) | 0.0231 | 0.0009 | 0.0001 | 0.0000 | 0.0037 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9878 | 0.9931 | 0.9932 | 0.9921 | 0.9990 | 0.9998 | 0.9999 | 0.9999 | |
\(\hat {s}\) | 0.0298 | 0.0106 | 0.0064 | 0.0051 | 0.0062 | 0.0007 | 0.0002 | 0.0001 | |
α= 0.2 | \(\hat {\alpha }\) | 0.0343 | 0.0019 | 0.0001 | 0.0000 | 0.0055 | 0.0000 | 0.0000 | 0.0000 |
\(\hat {\rho }\) | 0.9835 | 0.9906 | 0.9912 | 0.9903 | 0.9985 | 0.9998 | 0.9999 | 0.9999 | |
\(\hat {s}\) | 0.0357 | 0.0131 | 0.0078 | 0.0059 | 0.0080 | 0.0010 | 0.0003 | 0.0002 | |
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 | |
\(\hat {s}\) | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | |
α= 0.1 | \(\hat {\alpha }\) | 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= 0.2 | \(\hat {\alpha }\) | 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 |