Table 6 Summary of MLE’s results simulation of the E-X\(^{\theta }\)E for some values of \(\alpha\) and \(\theta\).
Value | n | \(\mid\)Bias\(\mid\) | RMSE | Variance | Value | n | \(\mid\)bias\(\mid\) | RMSE | Variance | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\alpha\) | \(\theta\) | \({\hat{\alpha }}\) | \({\hat{\theta }}\) | \(\alpha\) | \(\theta\) | \(\alpha\) | \(\theta\) | \(\alpha\) | \(\theta\) | \({\hat{\alpha }}\) | \({\hat{\theta }}\) | \(\alpha\) | \(\theta\) | \(\alpha\) | \(\theta\) | ||
2 | 0.1 | 30 | 0.193 | 0.009 | 0.884 | 0.03 | 0.744 | 8.1 \(\times 10^{-4}\) | 0.3 | 0.2 | 30 | 0.025 | 0.020 | 0.134 | 0.053 | 0.015 | 2.5 \(\times 10^{-3}\) |
45 | 0.111 | 0.004 | 0.617 | 0.019 | 0.368 | 3.4 \(\times 10^{-4}\) | 45 | 0.013 | 0.013 | 0.093 | 0.040 | 8.5 \(\times 10^{-3}\) | 1.5 \(\times 10^{-3}\) | ||||
60 | 0.081 | 0.004 | 0.516 | 0.015 | 0.261 | 2.1 \(\times 10^{-4}\) | 60 | 0.009 | 0.009 | 0.076 | 0.034 | 5.5 \(\times 10^{-3}\) | 1.1 \(\times 10^{-3}\) | ||||
90 | 0.038 | 0.003 | 0.403 | 0.012 | 0.161 | 1.36 \(\times 10^{-4}\) | 90 | 0.007 | 0.005 | 0.06 | 0.024 | 3.5 \(\times 10^{-3}\) | 5.7 \(\times 10^{-4}\) | ||||
120 | 0.048 | 0.002 | 0.359 | 0.01 | 0.127 | 9.9 \(\times 10^{-5}\) | 120 | 0.004 | 0.004 | 0.051 | 0.021 | 2.6 \(\times 10^{-3}\) | 4.2 \(\times 10^{-4}\) | ||||
150 | 0.04 | 0.001 | 0.310 | 0.009 | 0.094 | 7.5 \(\times 10^{-5}\) | 150 | 0.006 | 0.003 | 0.046 | 0.018 | 2.1 \(\times 10^{-3}\) | 3.2 \(\times 10^{-4}\) | ||||
200 | 0.009 | 0.001 | 0.265 | 0.008 | 0.070 | 5.5 \(\times 10^{-5}\) | 200 | 0.002 | 0.002 | 0.038 | 0.016 | 1.5 \(\times 10^{-4}\) | 2.5 \(\times 10^{-4}\) | ||||
0.1 | 2 | 30 | 0.0004 | 0.135 | 0.048 | 0.456 | 2.3 \(\times 10^{-3}\) | 0.19 | 0.6 | 0.5 | 30 | 0.077 | 0.055 | 0.515 | 0.124 | 0.259 | 0.013 |
45 | 0.0002 | 0.073 | 0.037 | 0.353 | 1.4 \(\times 10^{-3}\) | 0.112 | 45 | 0.038 | 0.032 | 0.25 | 0.093 | 0.062 | 7.8 \(\times 10^{-3}\) | ||||
60 | 0.0004 | 0.050 | 0.032 | 0.300 | 1.0 \(\times 10^{-3}\) | 0.085 | 60 | 0.02 | 0.021 | 0.17 | 0.070 | 0.031 | 5.2 \(\times 10^{-3}\) | ||||
90 | 0.0006 | 0.044 | 0.025 | 0.230 | 6.4 \(\times 10^{-4}\) | 0.051 | 90 | 0.18 | 0.011 | 0.145 | 0.057 | 0.021 | 3.1 \(\times 10^{-3}\) | ||||
120 | 0.0002 | 0.033 | 0.022 | 0.190 | 4.8 \(\times 10^{-4}\) | 0.035 | 120 | 0.009 | 0.009 | 0.112 | 0.049 | 0.012 | 2.3 \(\times 10^{-3}\) | ||||
150 | 0.0007 | 0.017 | 0.019 | 0.16 | 3.8 \(\times 10^{-4}\) | 0.027 | 150 | 0.014 | 0.007 | 0.102 | 0.043 | 0.010 | 1.7 \(\times 10^{-3}\) | ||||
200 | 0.0006 | 0.024 | 0.017 | 0.15 | 2.8 \(\times 10^{-4}\) | 0.022 | 200 | 0.0037 | 0.006 | 0.088 | 0.037 | 7.7 \(\times 10^{-3}\) | 1.4 \(\times 10^{-4}\) | ||||
0.5 | 1.5 | 30 | 0.026 | 0.102 | 0.362 | 0.311 | 0.13 | 0.087 | 0.1 | 0.2 | 30 | 0.003 | 0.022 | 0.034 | 0.057 | 1.2 \(\times 10^{-3}\) | 2.8 \(\times 10^{-3}\) |
45 | 0.0047 | 0.077 | 0.161 | 0.231 | 0.026 | 0.047 | 45 | 0.002 | 0.012 | 0.026 | 0.04 | 6.7 \(\times 10^{-4}\) | 1.5 \(\times 10^{-3}\) | ||||
60 | 0.0087 | 0.079 | 0.154 | 0.243 | 0.024 | 0.035 | 60 | 0.002 | 0.008 | 0.023 | 0.031 | 5.4 \(\times 10^{-4}\) | 9.1 \(\times 10^{-4}\) | ||||
90 | 0.0053 | 0.033 | 0.104 | 0.153 | 0.011 | 0.022 | 90 | 0.0002 | 0.007 | 0.019 | 0.026 | 3.6 \(\times 10^{-4}\) | 6.5 \(\times 10^{-4}\) | ||||
120 | 0.0055 | 0.022 | 0.087 | 0.129 | 7.5 \(\times 10^{-3}\) | 0.016 | 120 | 0.0001 | 0.004 | 0.016 | 0.02 | 2.5 \(\times 10^{-4}\) | 3.9 \(\times 10^{-4}\) | ||||
150 | 0.0002 | 0.018 | 0.078 | 0.112 | 6 \(\times 10^{-3}\) | 0.012 | 150 | 0.00007 | 0.004 | 0.014 | 0.019 | 1.9 \(\times 10^{-4}\) | 3.5 \(\times 10^{-4}\) | ||||
200 | 0.0005 | 0.010 | 0.068 | 0.101 | 4.6 \(\times 10^{-3}\) | 0.010 | 200 | 0.00003 | 0.002 | 0.012 | 0.015 | 1.3 \(\times 10^{-4}\) | 2.3 \(\times 10^{-4}\) | ||||
1.2 | 1.5 | 30 | 0.491 | 0.125 | 2.054 | 0.303 | 3.97 | 0.077 | 1 | 2 | 30 | 0.153 | 0.131 | 0787 | 0.382 | 0.596 | 0.129 |
45 | 0.189 | 0.074 | 0.895 | 0.238 | 0.765 | 0.051 | 45 | 0.089 | 0.103 | 0.524 | 0.3 | 0.265 | 0.079 | ||||
60 | 0.234 | 0.062 | 0.592 | 0.191 | 0.339 | 0.032 | 60 | 0.047 | 0.068 | 0.366 | 0.233 | 0.132 | 0.050 | ||||
90 | 0.087 | 0.037 | 0.407 | 0.149 | 0.158 | 0.021 | 90 | 0.036 | 0.043 | 0.262 | 0.187 | 0.067 | 0.033 | ||||
120 | 0.056 | 0.028 | 0.33 | 0.128 | 0.107 | 0.016 | 120 | 0.03 | 0.028 | 0.213 | 0.159 | 0.045 | 0.024 | ||||
150 | 0.035 | 0.024 | 0.265 | 0.110 | 0.069 | 0.012 | 150 | 0.018 | 0.026 | 0.196 | 0.145 | 0.038 | 0.020 | ||||
200 | 0.014 | 0.017 | 0.208 | 0.091 | 0.043 | 7.9 \(\times 10^{-3}\) | 200 | 0.011 | 0.020 | 0.168 | 0.125 | 0.028 | 0.015 | ||||
1 | 0.2 | 30 | 0.159 | 0.022 | 0.573 | 0.061 | 0.303 | 3.2\(\times 10^{-3}\) | 0.2 | 1 | 30 | 0.001 | 0.084 | 0.076 | 0.235 | 5.7 \(\times 10^{-3}\) | 0.048 |
45 | 0.081 | 0.011 | 0.506 | 0.04 | 0.249 | 1.5 \(\times 10^{-3}\) | 45 | 0.003 | 0.058 | 0.062 | 0.177 | 3.8 \(\times 10^{-3}\) | 0.028 | ||||
60 | 0.046 | 0.009 | 0.274 | 0.032 | 0.073 | 9.5 \(\times 10^{-4}\) | 60 | 0.004 | 0.048 | 0.05 | 0.149 | 2.5 \(\times 10^{-3}\) | 0.020 | ||||
90 | 0.033 | 0.004 | 0.218 | 0.025 | 0.046 | 5.9 \(\times 10^{-4}\) | 90 | 0.003 | 0.033 | 0.042 | 0.124 | 1.7 \(\times 10^{-3}\) | 0.014 | ||||
120 | 0.034 | 0.006 | 0.196 | 0.021 | 0.037 | 4.2 \(\times 10^{-4}\) | 120 | 0.0001 | 0.021 | 0.037 | 0.099 | 1.4 \(\times 10^{-3}\) | 9.2\(\times 10^{-3}\) | ||||
150 | 0.018 | 0.003 | 0.171 | 0.018 | 0.029 | 3.3 \(\times 10^{-4}\) | 150 | 0.0021 | 0.019 | 0.031 | 0.088 | 9.8\(\times 10^{-4}\) | 7.3\(\times 10^{-4}\) | ||||
200 | 0.016 | 0.001 | 0.145 | 0.016 | 0.021 | 2.4 \(\times 10^{-4}\) | 200 | 0.0016 | 0.009 | 0.03 | 0.073 | 8.8 \(\times 10^{-4}\) | 5.3 \(\times 10^{-4}\) | ||||
