Table 12 Results confirming the necessary condition for Theorem 2.
Estimator | Condition | \(\lambda _1\) | \(\lambda _2\) | \(\lambda _3\) | \(\lambda _4\) | \(\lambda _5\) | \(\lambda _6\) | \(\lambda _7\) | |
|---|---|---|---|---|---|---|---|---|---|
GPRRE | \(\hat{k}^*_{1 }\) | \(\hat{\varphi }k^*_{1} + 2\hat{\varphi }\lambda - k^*_{1}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.26E+05 | 2.46E+05 | 2.17E+04 | 8.61E+03 | 6.44E+02 |
\(\hat{k}^*_{2 }\) | \(\hat{\varphi }k^*_{2} + 2\hat{\varphi }\lambda - k^*_{2}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.26E+05 | 2.45E+05 | 2.15E+04 | 7.52E+03 | 7.19E+02 | |
\(\hat{k}^*_{3 }\) | \(\hat{\varphi }k^*_{3} + 2\hat{\varphi }\lambda - k^*_{3}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.26E+05 | 2.45E+05 | 2.15E+04 | 7.52E+03 | 7.19E+02 | |
\(\hat{k}^*_{4 }\) | \(\hat{\varphi }k^*_{4} + 2\hat{\varphi }\lambda - k^*_{4}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.51E+05 | 2.25E+04 | 1.41E+04 | 2.69E+02 | |
\(\hat{k}^*_{5 }\) | \(\hat{\varphi }k^*_{5} + 2\hat{\varphi }\lambda - k^*_{5}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.49E+05 | 2.22E+04 | 1.22E+04 | 3.98E+02 | |
\(\hat{k}^*_{6 }\) | \(\hat{\varphi }k^*_{6} + 2\hat{\varphi }\lambda - k^*_{6}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.49E+05 | 2.22E+04 | 1.22E+04 | 4.00E+02 | |
\(\hat{k}^*_{7 }\) | \(\hat{\varphi }k^*_{7} + 2\hat{\varphi }\lambda - k^*_{7}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.50E+05 | 2.25E+04 | 1.40E+04 | 2.74E+02 | |
\(\hat{k}^*_{8 }\) | \(\hat{\varphi }k^*_{8} + 2\hat{\varphi }\lambda - k^*_{8}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.27E+05 | 2.47E+05 | 2.20E+04 | 1.06E+04 | 5.07E+02 | |
\(\hat{k}^*_{9 }\) | \(\hat{\varphi }k^*_{9} + 2\hat{\varphi }\lambda - k^*_{9}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.27E+05 | 2.48E+05 | 2.21E+04 | 1.15E+04 | 4.44E+02 | |
\(\hat{k}^*_{10}\) | \(\hat{\varphi }k^*_{10} + 2\hat{\varphi }\lambda - k^*_{10}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.51E+05 | 2.25E+04 | 1.41E+04 | 2.67E+02 | |
\(\hat{k}^*_{11}\) | \(\hat{\varphi }k^*_{11} + 2\hat{\varphi }\lambda - k^*_{11}\lambda \alpha _j^2\) | 5.54E+09 | 2.31E+07 | 5.27E+05 | 2.50E+05 | 2.24E+04 | 1.35E+04 | 3.09E+02 | |
\(\hat{k}^*_{12}\) | \(\hat{\varphi }k^*_{12} + 2\hat{\varphi }\lambda - k^*_{12}\lambda \alpha _j^2\) | 5.53E+09 | 2.29E+07 | 5.26E+05 | 2.41E+05 | 2.08E+04 | 3.41E+03 | 1.00E+03 | |
\(\hat{k}^*_{13}\) | \(\hat{\varphi }k^*_{13} + 2\hat{\varphi }\lambda - k^*_{13}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.26E+05 | 2.43E+05 | 2.12E+04 | 5.71E+03 | 8.43E+02 | |
\(\hat{k}^*_{14}\) | \(\hat{\varphi }k^*_{14} + 2\hat{\varphi }\lambda - k^*_{14}\lambda \alpha _j^2\) | 5.53E+09 | 2.29E+07 | 5.25E+05 | 2.39E+05 | 2.05E+04 | 1.51E+03 | 1.13E+03 | |
\(\hat{k}^*_{15}\) | \(\hat{\varphi }k^*_{15} + 2\hat{\varphi }\lambda - k^*_{15}\lambda \alpha _j^2\) | 5.53E+09 | 2.30E+07 | 5.26E+05 | 2.44E+05 | 2.13E+04 | 6.33E+03 | 8.00E+02 | |
