Research Article
Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation
Table 1
The results of MLEs and Bayesian estimates.
| | MLE | Bayesian | | | | | | | | |
| , | n = 30 | ABias | 0.020 | 0.015 | 0.015 | 0.010 | 0.016 | 0.012 | 0.008 | 0.012 | MSE | 0.003 | 0.004 | 0.004 | 0.004 | 0.003 | 0.004 | 0.004 | 0.003 | CW (CL) | 0.214 | 0.248 | 0.248 | 0.248 | 0.213 | 0.243 | 0.239 | 0.230 | CP | 0.950 | 0.938 | 0.957 | 0.948 | 0.943 | 0.932 | 0.949 | 0.951 | n = 50 | ABias | 0.011 | 0.009 | 0.008 | 0.007 | 0.010 | 0.009 | 0.004 | 0.009 | MSE | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | 0.002 | CW (CL) | 0.176 | 0.176 | 0.176 | 0.176 | 0.162 | 0.188 | 0.185 | 0.179 | CP | 0.954 | 0.940 | 0.938 | 0.951 | 0.956 | 0.943 | 0.949 | 0.942 | n = 70 | ABias | 0.010 | 0.006 | 0.006 | 0.006 | 0.008 | 0.006 | 0.004 | 0.007 | MSE | 0.001 | 0.002 | 0.002 | 0.001 | 0.001 | 0.002 | 0.002 | 0.001 | CW (CL) | 0.124 | 0.176 | 0.176 | 0.124 | 0.136 | 0.157 | 0.156 | 0.143 | CP | 0.952 | 0.957 | 0.946 | 0.951 | 0.958 | 0.945 | 0.949 | 0.948 |
| , | n = 30 | ABias | 0.032 | 0.048 | 0.046 | 0.032 | 0.028 | 0.041 | 0.027 | 0.038 | MSE | 0.011 | 0.034 | 0.034 | 0.026 | 0.011 | 0.031 | 0.030 | 0.025 | CW (CL) | 0.392 | 0.702 | 0.702 | 0.620 | 0.394 | 0.664 | 0.659 | 0.587 | CP | 0.947 | 0.945 | 0.956 | 0.951 | 0.942 | 0.930 | 0.945 | 0.951 | n = 50 | ABias | 0.018 | 0.028 | 0.026 | 0.022 | 0.020 | 0.027 | 0.026 | 0.025 | MSE | 0.006 | 0.018 | 0.018 | 0.014 | 0.006 | 0.017 | 0.018 | 0.014 | CW (CL) | 0.304 | 0.512 | 0.512 | 0.464 | 0.299 | 0.501 | 0.504 | 0.448 | CP | 0.948 | 0.944 | 0.945 | 0.957 | 0.954 | 0.949 | 0.946 | 0.937 | n = 70 | ABias | 0.017 | 0.019 | 0.021 | 0.019 | 0.013 | 0.021 | 0.017 | 0.019 | MSE | 0.004 | 0.012 | 0.012 | 0.010 | 0.004 | 0.012 | 0.012 | 0.010 | CW (CL) | 0.248 | 0.430 | 0.430 | 0.392 | 0.250 | 0.417 | 0.418 | 0.375 | CP | 0.945 | 0.958 | 0.948 | 0.950 | 0.951 | 0.929 | 0.951 | 0.946 |
| , | n = 30 | ABias | 0.045 | 0.088 | 0.086 | 0.066 | 0.037 | 0.072 | 0.073 | 0.060 | MSE | 0.022 | 0.101 | 0.099 | 0.076 | 0.022 | 0.090 | 0.091 | 0.068 | CW (CL) | 0.554 | 1.196 | 1.188 | 1.052 | 0.557 | 1.129 | 1.133 | 0.979 | CP | 0.941 | 0.951 | 0.954 | 0.946 | 0.940 | 0.939 | 0.950 | 0.951 | n = 50 | ABias | 0.026 | 0.051 | 0.049 | 0.040 | 0.026 | 0.059 | 0.049 | 0.040 | MSE | 0.013 | 0.052 | 0.051 | 0.040 | 0.013 | 0.052 | 0.050 | 0.038 | CW (CL) | 0.430 | 0.868 | 0.868 | 0.764 | 0.425 | 0.854 | 0.850 | 0.744 | CP | 0.950 | 0.944 | 0.941 | 0.956 | 0.956 | 0.948 | 0.948 | 0.940 | n = 70 | ABias | 0.024 | 0.036 | 0.039 | 0.035 | 0.019 | 0.037 | 0.030 | 0.030 | MSE | 0.009 | 0.034 | 0.036 | 0.028 | 0.009 | 0.034 | 0.033 | 0.026 | CW (CL) | 0.350 | 0.712 | 0.722 | 0.644 | 0.358 | 0.703 | 0.703 | 0.619 | CP | 0.944 | 0.956 | 0.949 | 0.949 | 0.958 | 0.940 | 0.952 | 0.946 |
| , | n = 30 | ABias | 0.068 | 0.142 | 0.158 | 0.307 | 0.023 | 0.120 | 0.162 | 0.137 | MSE | 0.062 | 0.391 | 0.442 | 0.913 | 0.059 | 0.355 | 0.447 | 0.664 | CW (CL) | 0.936 | 2.388 | 2.532 | 3.548 | 0.924 | 2.207 | 2.417 | 3.063 | CP | 0.950 | 0.940 | 0.950 | 0.950 | 0.940 | 0.950 | 0.930 | 0.960 | n = 50 | ABias | 0.046 | 0.099 | 0.111 | 0.191 | 0.018 | 0.076 | 0.103 | 0.091 | MSE | 0.035 | 0.203 | 0.231 | 0.444 | 0.034 | 0.189 | 0.219 | 0.369 | CW (CL) | 0.712 | 1.722 | 1.834 | 2.504 | 0.703 | 1.633 | 1.738 | 2.043 | CP | 0.960 | 0.950 | 0.950 | 0.960 | 0.960 | 0.920 | 0.950 | 0.970 | n = 70 | ABias | 0.039 | 0.068 | 0.082 | 0.157 | 0.013 | 0.068 | 0.051 | 0.090 | MSE | 0.026 | 0.134 | 0.153 | 0.301 | 0.023 | 0.131 | 0.139 | 0.258 | CW (CL) | 0.608 | 1.408 | 1.498 | 1.627 | 0.586 | 1.364 | 1.416 | 1.941 | CP | 0.930 | 0.940 | 0.940 | 0.960 | 0.910 | 0.940 | 0.950 | 0.960 |
| , | n = 30 | ABias | 0.094 | 0.296 | 0.295 | 0.237 | 0.060 | 0.264 | 0.247 | 0.147 | MSE | 0.099 | 0.890 | 0.892 | 0.680 | 0.094 | 0.811 | 0.790 | 0.519 | CW (CL) | 1.176 | 3.510 | 3.518 | 3.096 | 1.166 | 3.312 | 3.237 | 2.723 | CP | 0.943 | 0.949 | 0.950 | 0.952 | 0.949 | 0.931 | 0.948 | 0.952 | n = 50 | ABias | 0.055 | 0.169 | 0.166 | 0.136 | 0.040 | 0.185 | 0.169 | 0.101 | MSE | 0.055 | 0.427 | 0.428 | 0.326 | 0.055 | 0.432 | 0.418 | 0.290 | CW (CL) | 1.176 | 3.510 | 3.518 | 3.096 | 0.898 | 2.443 | 2.388 | 2.050 | CP | 0.953 | 0.946 | 0.941 | 0.961 | 0.952 | 0.946 | 0.945 | 0.952 | n = 70 | ABias | 0.051 | 0.123 | 0.119 | 0.051 | 0.031 | 0.129 | 0.101 | 0.063 | MSE | 0.040 | 0.281 | 0.285 | 0.226 | 0.038 | 0.277 | 0.263 | 0.195 | CW (CL) | 0.754 | 2.022 | 2.030 | 1.804 | 0.753 | 1.979 | 1.948 | 1.695 | CP | 0.942 | 0.953 | 0.945 | 0.942 | 0.961 | 0.942 | 0.949 | 0.942 |
| .5, | n = 30 | ABias | 0.086 | 0.383 | 0.493 | 0.507 | 0.037 | 0.268 | 0.426 | 0.178 | MSE | 0.089 | 1.832 | 2.583 | 2.476 | 0.219 | 1.497 | 2.411 | 0.908 | CW (CL) | 1.122 | 5.088 | 5.996 | 5.840 | 0.843 | 4.543 | 5.563 | 3.387 | CP | 0.950 | 0.950 | 0.950 | 0.950 | 0.942 | 0.950 | 0.880 | 0.938 | n = 50 | ABias | 0.053 | 0.226 | 0.279 | 0.297 | 0.019 | 0.197 | 0.304 | 0.095 | MSE | 0.051 | 0.870 | 1.194 | 1.132 | 0.049 | 0.815 | 1.197 | 0.865 | CW (CL) | 0.858 | 3.548 | 4.141 | 4.006 | 0.842 | 3.376 | 3.996 | 3.541 | CP | 0.950 | 0.940 | 0.940 | 0.970 | 0.930 | 0.910 | 0.930 | 0.950 | n = 70 | ABias | 0.049 | 0.173 | 0.236 | 0.244 | 0.014 | 0.166 | 0.149 | 0.091 | MSE | 0.036 | 0.574 | 0.800 | 0.766 | 0.033 | 0.543 | 0.688 | 0.607 | CW (CL) | 0.712 | 2.894 | 3.382 | 3.294 | 0.697 | 2.770 | 3.116 | 2.997 | CP | 0.940 | 0.940 | 0.950 | 0.960 | 0.950 | 0.930 | 0.930 | 0.960 |
| , | n = 30 | ABias | 0.111 | 0.390 | 0.489 | 0.326 | 0.067 | 0.374 | 0.418 | 0.168 | MSE | 0.140 | 1.535 | 2.191 | 1.234 | 0.134 | 1.435 | 1.938 | 0.891 | CW (CL) | 1.402 | 4.610 | 5.476 | 4.164 | 1.395 | 4.375 | 5.016 | 3.545 | CP | 0.943 | 0.950 | 0.954 | 0.947 | 0.946 | 0.903 | 0.915 | 0.955 | n = 50 | ABias | 0.068 | 0.232 | 0.279 | 0.183 | 0.041 | 0.236 | 0.262 | 0.108 | MSE | 0.079 | 0.737 | 1.017 | 0.577 | 0.078 | 0.725 | 0.977 | 0.496 | CW (CL) | 1.066 | 3.240 | 3.798 | 2.892 | 1.074 | 3.167 | 3.657 | 2.697 | CP | 0.950 | 0.940 | 0.949 | 0.961 | 0.949 | 0.917 | 0.886 | 0.952 | n = 70 | ABias | 0.060 | 0.164 | 0.214 | 0.160 | 0.028 | 0.166 | 0.166 | 0.069 | MSE | 0.057 | 0.479 | 0.669 | 0.399 | 0.037 | 0.472 | 0.622 | 0.333 | CW (CL) | 0.902 | 2.636 | 3.094 | 2.394 | 0.903 | 2.581 | 2.972 | 2.217 | CP | 0.948 | 0.950 | 0.950 | 0.943 | 0.966 | 0.887 | 0.883 | 0.950 |
| , | n = 30 | ABias | 0.152 | 0.717 | 0.607 | 0.533 | 0.068 | 0.663 | 0.536 | 0.115 | MSE | 0.257 | 4.574 | 3.552 | 2.888 | 0.237 | 4.168 | 3.127 | 1.682 | CW (CL) | 1.896 | 7.898 | 6.994 | 6.322 | 1.873 | 7.403 | 6.356 | 4.974 | CP | 0.938 | 0.949 | 0.949 | 0.954 | 0.836 | 0.928 | 0.955 | 0.956 | n = 50 | ABias | 0.091 | 0.414 | 0.352 | 0.289 | 0.053 | 0.438 | 0.357 | 0.133 | MSE | 0.143 | 2.075 | 1.637 | 1.277 | 0.141 | 2.080 | 1.606 | 1.038 | CW (CL) | 1.440 | 5.410 | 4.822 | 4.282 | 1.448 | 5.311 | 4.641 | 3.905 | CP | 0.952 | 0.947 | 0.934 | 0.966 | 0.761 | 0.919 | 0.932 | 0.963 | n = 70 | ABias | 0.082 | 0.309 | 0.268 | 0.259 | 0.031 | 0.283 | 0.232 | 0.060 | MSE | 0.103 | 1.352 | 1.066 | 0.883 | 0.099 | 1.297 | 1.012 | 0.701 | CW (CL) | 1.214 | 4.394 | 3.908 | 3.542 | 1.222 | 4.275 | 3.773 | 3.240 | CP | 0.940 | 0.951 | 0.945 | 0.953 | 0.696 | 0.928 | 0.957 | 0.953 |
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