A Compound Class of Inverse-Power Muth and Power Series Distributions
Abstract
:1. Introduction
2. The Model
2.1. Model Construction
2.2. Model Properties
2.3. Shannon Entropy
2.4. Pseudo-Random Number Generator of the Model
Algorithm 1 Simulating values of the IPM-PS model |
|
3. Parameter Estimation
EM Algorithm
- E step: For , define and calculate:
- step M-I: is updated as the solution of the non-linear equation
- step M-II: Given the vector , update by maximizingin relation to each of the parameters.
- If the convergence condition is reached, the algorithm stops. Otherwise, we return to step E for a new iteration.
4. Simulation Study
5. Applications
5.1. Application 1
5.2. Application 2
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Additional Plots
Appendix B. Explicit form for the Integral in u(r) of r-th Moment of the IPM- PS
Appendix C. Explicit form for the Integral in Equation (9) for the Shannon Entropy
Appendix D. Inverse and Derivatives of A(·)
Appendix E. Oakes’ Method for the IMP-PS Distribution
Appendix E.1. Double Derivatives of the Function Q in Relation to the Parameters ζ
Appendix E.2. Double Derivatives of Function Q in Relation to Parameters and Expectation Mi
Appendix F. PDF of Distributions Used in Applications
References
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Distribution | Notation | |||
---|---|---|---|---|
Geometric | Geo | 1 | ||
Poisson | Po | |||
Bell | Be | |||
Logarithmic | Lo |
IPM-G | |||||||||
---|---|---|---|---|---|---|---|---|---|
(0.1;−1) | (0.2;−1) | (0.5;−1) | (0.1;0.5) | (0.2;0.5) | (0.5;0.5) | (0.1;1) | (0.2;1) | (0.5;1) | |
1.4953 | 1.3291 | 0.8307 | 1.0860 | 0.9654 | 0.6034 | 0.9569 | 0.8505 | 0.5316 | |
3.7277 | 3.3135 | 2.0710 | 1.7405 | 1.5471 | 0.9670 | 1.0671 | 0.9485 | 0.5928 | |
V | 1.4919 | 1.5470 | 1.3809 | 0.5611 | 0.6152 | 0.6029 | 0.1515 | 0.2251 | 0.3102 |
IPM-L | |||||||||
1.5769 | 1.4891 | 1.1984 | 1.1453 | 1.0815 | 0.8705 | 1.0091 | 0.9529 | 0.7669 | |
3.9312 | 3.7123 | 2.9878 | 1.8355 | 1.7333 | 1.3950 | 1.1253 | 1.0627 | 0.8553 | |
V | 1.4447 | 1.4950 | 1.5515 | 0.5238 | 0.5636 | 0.6373 | 0.1070 | 0.1546 | 0.2671 |
IPM-P | |||||||||
(0.1;−1) | (0.5;−1) | (2;−1) | (0.1;0.5) | (0.5;0.5) | (2;0.5) | (0.1;1) | (0.5;1) | (2;1) | |
1.5797 | 1.2805 | 0.5201 | 1.1474 | 0.9301 | 0.3777 | 1.0109 | 0.8194 | 0.3328 | |
3.9383 | 3.1924 | 1.2966 | 1.8388 | 1.4906 | 0.6054 | 1.1273 | 0.9138 | 0.3711 | |
V | 1.4428 | 1.5527 | 1.0261 | 0.5224 | 0.6255 | 0.4627 | 0.1054 | 0.2423 | 0.2604 |
IPM-B | |||||||||
1.4981 | 0.9098 | 0.0056 | 1.0881 | 0.6608 | 0.0041 | 0.9587 | 0.5822 | 0.0036 | |
3.7348 | 2.2681 | 0.0139 | 1.7438 | 1.0590 | 0.0065 | 1.0691 | 0.6492 | 0.0040 | |
V | 1.4905 | 1.4404 | 0.0139 | 0.5599 | 0.6224 | 0.0065 | 0.1500 | 0.3103 | 0.0040 |
True Value | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
bias | RMSE | SE | CP | bias | RMSE | SE | CP | bias | RMSE | SE | CP | ||||
0.5 | 1 | 0.5 | −0.0783 | 0.2152 | 0.7088 | 0.9310 | −0.0516 | 0.1879 | 0.5608 | 0.9570 | −0.0270 | 0.1543 | 0.3481 | 0.9660 | |
0.1415 | 0.3005 | 0.3507 | 0.9830 | 0.0851 | 0.2266 | 0.3132 | 0.9770 | 0.0366 | 0.1552 | 0.2365 | 0.9600 | ||||
0.0542 | 0.1328 | 0.3841 | 0.9990 | 0.0315 | 0.0887 | 0.2806 | 0.9700 | 0.0115 | 0.0490 | 0.1764 | 0.9660 | ||||
−0.0336 | 0.1055 | 0.5604 | 0.9990 | −0.0156 | 0.0583 | 0.4467 | 0.9870 | −0.0012 | 0.0152 | 0.2789 | 0.9700 | ||||
0.2 | 0.0432 | 0.2111 | 0.7731 | 0.9770 | 0.0404 | 0.1643 | 0.5337 | 0.9710 | 0.0244 | 0.1133 | 0.4813 | 0.9650 | |||
0.0291 | 0.2075 | 0.3240 | 0.9870 | 0.0036 | 0.1581 | 0.2447 | 0.9710 | −0.0144 | 0.1000 | 0.2726 | 0.9660 | ||||
0.0717 | 0.1825 | 0.3119 | 0.9800 | 0.0427 | 0.1368 | 0.2091 | 0.9770 | 0.0049 | 0.0633 | 0.1412 | 0.9660 | ||||
−0.0824 | 0.1625 | 0.7174 | 0.9800 | −0.0530 | 0.1189 | 0.5045 | 0.9770 | −0.0118 | 0.0438 | 0.4565 | 0.9600 | ||||
0.1 | 0.0318 | 0.3060 | 0.7434 | 0.9910 | 0.0370 | 0.1503 | 0.5364 | 0.9820 | 0.0224 | 0.0873 | 0.3129 | 0.9740 | |||
0.0220 | 0.1894 | 0.3427 | 0.9900 | −0.0004 | 0.1342 | 0.1942 | 0.9820 | −0.0149 | 0.0793 | 0.1091 | 0.9690 | ||||
0.1113 | 0.2970 | 0.2888 | 0.9950 | 0.0495 | 0.1596 | 0.2069 | 0.9910 | 0.0049 | 0.0714 | 0.1289 | 0.9790 | ||||
−0.1054 | 0.1927 | 0.8066 | 0.9850 | −0.0620 | 0.1356 | 0.5439 | 0.9810 | −0.0157 | 0.0545 | 0.3312 | 0.9740 | ||||
−0.2 | −0.0930 | 0.5089 | 0.9152 | 0.9920 | −0.3430 | 0.3655 | 0.6773 | 0.8670 | −0.0577 | 0.1943 | 0.4614 | 0.9770 | |||
−0.0131 | 0.1356 | 0.5921 | 0.9820 | −0.0122 | 0.0952 | 0.4580 | 0.9770 | −0.0102 | 0.0600 | 0.1467 | 0.9690 | ||||
0.1882 | 0.5432 | 0.3252 | 0.9870 | 0.6742 | 0.5457 | 0.2439 | 0.9790 | 0.0299 | 0.1385 | 0.1670 | 0.9670 | ||||
−0.0843 | 0.2032 | 1.5664 | 0.9870 | −0.0321 | 0.1230 | 1.2458 | 0.9790 | −0.0027 | 0.0490 | 0.6344 | 0.9690 | ||||
2 | 0.5 | −0.0806 | 0.2093 | 0.6795 | 0.9480 | −0.0514 | 0.1836 | 0.5117 | 0.9580 | −0.0259 | 0.1457 | 0.3659 | 0.9640 | ||
0.1295 | 0.3017 | 0.3595 | 0.9480 | 0.0793 | 0.2278 | 0.2954 | 0.9570 | 0.0306 | 0.1502 | 0.2486 | 0.9610 | ||||
0.0753 | 0.3383 | 0.3635 | 0.9810 | 0.0493 | 0.2218 | 0.2676 | 0.9770 | 0.0130 | 0.1080 | 0.1768 | 0.9640 | ||||
−0.0298 | 0.1464 | 1.0769 | 0.9710 | −0.0147 | 0.0859 | 0.8224 | 0.9680 | −0.0026 | 0.0273 | 0.5823 | 0.9610 | ||||
0.2 | 0.0740 | 0.2136 | 0.7317 | 0.9840 | 0.0601 | 0.1733 | 0.5319 | 0.9720 | 0.0357 | 0.1092 | 0.3357 | 0.9620 | |||
−0.0043 | 0.2099 | 0.3093 | 0.9940 | −0.0331 | 0.1597 | 0.2420 | 0.9800 | −0.0301 | 0.0960 | 0.1545 | 0.9773 | ||||
0.1360 | 0.4811 | 0.2876 | 0.9990 | 0.0386 | 0.3577 | 0.2110 | 0.9840 | 0.0117 | 0.1953 | 0.1302 | 0.9730 | ||||
−0.1161 | 0.2187 | 1.3044 | 0.9890 | −0.0635 | 0.1622 | 1.0014 | 0.9810 | −0.0307 | 0.0906 | 0.6472 | 0.9790 | ||||
0.1 | 0.0607 | 0.2558 | 0.7508 | 0.9820 | 0.0583 | 0.1669 | 0.5352 | 0.9750 | 0.0319 | 0.0834 | 0.3194 | 0.9610 | |||
−0.0236 | 0.1883 | 0.2961 | 0.9880 | −0.0389 | 0.1417 | 0.1934 | 0.9850 | −0.0369 | 0.0802 | 0.1125 | 0.9770 | ||||
0.1271 | 0.5441 | 0.2831 | 0.9900 | 0.0454 | 0.3748 | 0.2013 | 0.9850 | −0.0184 | 0.2258 | 0.1271 | 0.9710 | ||||
−0.1214 | 0.2400 | 1.4822 | 0.9900 | −0.0758 | 0.1800 | 1.0671 | 0.9850 | −0.0251 | 0.1043 | 0.6699 | 0.9700 |
True Value | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
bias | RMSE | SE | CP | bias | RMSE | SE | CP | bias | RMSE | SE | CP | ||||
0.5 | 1 | 0.5 | −0.0622 | 0.2006 | 1.9905 | 0.9560 | −0.0363 | 0.1847 | 1.6212 | 0.9370 | −0.0024 | 0.1493 | 1.2423 | 0.9250 | |
0.0651 | 0.2493 | 0.3158 | 0.9810 | 0.0375 | 0.1969 | 0.2600 | 0.9750 | 0.0091 | 0.1452 | 0.2192 | 0.9640 | ||||
−0.0147 | 0.1516 | 0.3144 | 0.9950 | −0.0107 | 0.1141 | 0.2390 | 0.9880 | −0.0007 | 0.0612 | 0.1604 | 0.9600 | ||||
0.2062 | 0.6869 | 0.3978 | 0.9970 | 0.1358 | 0.4748 | 0.3247 | 0.9860 | 0.0308 | 0.2132 | 0.2472 | 0.9790 | ||||
1 | 1 | 0.5 | −0.0671 | 0.2085 | 1.9345 | 0.9780 | −0.0362 | 0.1757 | 1.5427 | 0.9680 | −0.0047 | 0.1427 | 1.1113 | 0.9630 | |
0.1230 | 0.2947 | 0.3180 | 0.9720 | 0.0747 | 0.2279 | 0.2599 | 0.9670 | 0.0300 | 0.1518 | 0.2044 | 0.9610 | ||||
0.0477 | 0.1797 | 0.3475 | 0.9990 | 0.0318 | 0.1319 | 0.2552 | 0.9890 | 0.0232 | 0.0788 | 0.1712 | 0.9750 | ||||
−0.0689 | 0.7308 | 0.3882 | 0.9860 | −0.0660 | 0.5294 | 0.3077 | 0.9800 | −0.0733 | 0.2831 | 0.2201 | 0.9700 | ||||
2 | 1 | 0.5 | −0.0498 | 0.2110 | 1.9715 | 0.9890 | −0.0269 | 0.1836 | 1.4665 | 0.9790 | 0.0140 | 0.1499 | 0.9608 | 0.9620 | |
0.2355 | 0.4180 | 0.3275 | 0.9850 | 0.1689 | 0.3203 | 0.2734 | 0.9770 | 0.0789 | 0.2077 | 0.2061 | 0.9660 | ||||
0.1628 | 0.2579 | 0.4444 | 0.9910 | 0.1314 | 0.2060 | 0.3156 | 0.9800 | 0.0900 | 0.1405 | 0.2010 | 0.9690 | ||||
−0.6167 | 1.0444 | 0.3925 | 0.9870 | −0.5380 | 0.8661 | 0.2884 | 0.9700 | −0.4140 | 0.6041 | 0.1868 | 0.9680 | ||||
0.2 | 0.0313 | 0.2495 | 2.3864 | 0.9870 | 0.0317 | 0.1588 | 1.8656 | 0.9750 | 0.0166 | 0.0944 | 1.1976 | 0.9690 | |||
−0.0124 | 0.1935 | 0.3168 | 0.9740 | −0.0226 | 0.1509 | 0.2235 | 0.9640 | −0.0148 | 0.0916 | 0.1374 | 0.9600 | ||||
−0.0050 | 0.2085 | 0.2938 | 0.9920 | −0.0132 | 0.1564 | 0.2118 | 0.9840 | −0.0053 | 0.0944 | 0.1310 | 0.9710 | ||||
0.0839 | 0.7085 | 0.6032 | 0.9760 | 0.0707 | 0.6137 | 0.4598 | 0.9690 | 0.0243 | 0.3381 | 0.2975 | 0.9600 | ||||
0.1 | −0.0712 | 2.0289 | 0.5630 | 0.9980 | 0.0211 | 0.4026 | 1.8901 | 0.9860 | 0.0226 | 0.0769 | 1.2122 | 0.9750 | |||
−0.0311 | 0.1827 | 1.5422 | 0.9890 | −0.0362 | 0.1445 | 0.2628 | 0.9630 | −0.0223 | 0.0838 | 0.1108 | 0.9600 | ||||
0.1051 | 2.3623 | 0.2907 | 0.9860 | −0.0180 | 0.4423 | 0.2134 | 0.9720 | −0.0124 | 0.1098 | 0.1326 | 0.9630 | ||||
0.1186 | 0.8074 | 2.1261 | 0.9930 | 0.1449 | 0.7578 | 0.5784 | 0.9820 | 0.0339 | 0.4028 | 0.3221 | 0.9720 | ||||
−0.2 | −0.4583 | 9.7743 | 2.7164 | 0.9830 | −0.4739 | 12.7700 | 2.1530 | 0.9730 | −0.0805 | 0.3246 | 1.5611 | 0.9640 | |||
−0.0713 | 0.1657 | 0.8047 | 0.9920 | −0.0574 | 0.1282 | 0.5628 | 0.9700 | −0.0297 | 0.0823 | 0.2070 | 0.9640 | ||||
1.9332 | 44.2624 | 0.2751 | 0.9810 | 0.6835 | 19.8681 | 0.2140 | 0.9750 | 0.0041 | 0.3105 | 0.1543 | 0.9670 | ||||
0.4587 | 1.1872 | 1.5392 | 0.9350 | 0.3772 | 0.9874 | 1.1873 | 0.9480 | 0.2001 | 0.6025 | 0.5607 | 0.9680 | ||||
2 | 0.5 | −0.0590 | 0.2046 | 1.9976 | 0.9990 | −0.0304 | 0.1798 | 1.6217 | 0.9970 | −0.0080 | 0.1471 | 1.2477 | 0.9780 | ||
0.0573 | 0.2482 | 0.3209 | 0.9720 | 0.0264 | 0.1917 | 0.2614 | 0.9660 | 0.0030 | 0.1434 | 0.2185 | 0.9640 | ||||
−0.0344 | 0.3092 | 0.3113 | 0.9710 | −0.0322 | 0.2375 | 0.2369 | 0.9630 | −0.0191 | 0.1419 | 0.1599 | 0.9600 | ||||
0.1937 | 0.6670 | 0.7954 | 0.9920 | 0.1345 | 0.4885 | 0.6508 | 0.9910 | 0.0442 | 0.2692 | 0.4960 | 0.9960 | ||||
0.2 | 0.0340 | 0.2413 | 2.4225 | 0.9990 | 0.0386 | 0.1560 | 1.8048 | 0.9970 | 0.0244 | 0.1040 | 1.2031 | 0.9750 | |||
−0.0328 | 0.1934 | 0.3199 | 0.9900 | −0.0439 | 0.1442 | 0.2199 | 0.9800 | −0.0426 | 0.0997 | 0.1404 | 0.9750 | ||||
−0.0440 | 0.4214 | 0.2895 | 0.9760 | −0.0698 | 0.3155 | 0.2063 | 0.9660 | −0.0712 | 0.2201 | 0.1306 | 0.9600 | ||||
0.1083 | 0.7683 | 1.2248 | 0.9900 | 0.0881 | 0.6356 | 0.8844 | 0.9890 | 0.0750 | 0.4350 | 0.5925 | 0.9770 | ||||
0.1 | 0.0424 | 0.3420 | 2.5115 | 0.9950 | 0.0432 | 0.1536 | 1.8904 | 0.9840 | 0.0251 | 0.0815 | 1.2635 | 0.9760 | |||
−0.0544 | 0.1891 | 0.3697 | 0.9890 | −0.0615 | 0.1495 | 0.1969 | 0.9780 | −0.0453 | 0.0904 | 0.1155 | 0.9620 | ||||
−0.0891 | 0.5873 | 0.2878 | 0.9760 | −0.1124 | 0.3823 | 0.2082 | 0.9690 | −0.0837 | 0.2395 | 0.1376 | 0.9600 | ||||
0.1957 | 0.9359 | 1.4253 | 0.9770 | 0.1527 | 0.7494 | 0.9803 | 0.9700 | 0.0904 | 0.4559 | 0.6600 | 0.9600 |
True Value | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
bias | RMSE | SE | CP | bias | RMSE | SE | CP | bias | RMSE | SE | CP | ||||
0.5 | 1 | 0.5 | −0.0647 | 0.2052 | 0.8458 | 0.9460 | −0.0478 | 0.1802 | 0.6633 | 0.9590 | −0.0147 | 0.1416 | 0.4704 | 0.9680 | |
0.1279 | 0.2899 | 0.3174 | 0.9810 | 0.0847 | 0.2194 | 0.2624 | 0.9700 | 0.0290 | 0.1470 | 0.2165 | 0.9660 | ||||
0.0497 | 0.1415 | 0.3509 | 0.9860 | 0.0337 | 0.1005 | 0.2645 | 0.9770 | 0.0138 | 0.0524 | 0.1724 | 0.9660 | ||||
−0.0459 | 0.1832 | 0.4464 | 0.9800 | −0.02568 | 0.1092 | 0.3532 | 0.9700 | −0.0096 | 0.0355 | 0.2483 | 0.9660 | ||||
1 | 1 | 0.5 | −0.0847 | 0.2197 | 0.6442 | 0.9140 | −0.0610 | 0.1796 | 0.4419 | 0.9320 | −0.0410 | 0.1520 | 0.2637 | 0.9660 | |
0.3063 | 0.4903 | 0.3231 | 0.9410 | 0.1980 | 0.3386 | 0.2662 | 0.9530 | 0.1006 | 0.2180 | 0.2066 | 0.9650 | ||||
0.1920 | 0.2799 | 0.4858 | 0.9770 | 0.1303 | 0.1996 | 0.3501 | 0.9840 | 0.0621 | 0.1136 | 0.2256 | 0.9670 | ||||
−0.2241 | 0.3527 | 0.4201 | 0.9980 | −0.1380 | 0.2337 | 0.3036 | 0.9720 | −0.0570 | 0.1193& | 0.1887 | 0.9670 | ||||
2 | 1 | 0.5 | −0.1390 | 0.2193 | 0.3709 | 0.9800 | −0.1131 | 0.2028 | 0.2551 | 0.9720 | −0.0769 | 0.1726 | 0.1665 | 0.9690 | |
0.4175 | 0.7326 | 0.7068 | 0.9720 | 0.2426 | 0.3786 | 0.6131 | 0.9750 | 0.1299 | 0.2173 | 0.5371 | 0.9690 | ||||
0.2107 | 0.3243 | 1.1496 | 0.9710 | 0.1259 | 0.1832 | 0.8568 | 0.9860 | 0.0606 | 0.0907 | 0.6019 | 0.9970 | ||||
−0.1562 | 0.3359 | 0.4613 | 0.9520 | −0.0743 | 0.1487 | 0.3479 | 0.9600 | −0.0249 | 0.0592 | 0.2285 | 0.9670 | ||||
0.2 | 0.0480 | 0.2630 | 0.9211 | 0.9830 | 0.0406 | 0.1583 | 0.6999 | 0.9710 | 0.0302 | 0.1154 | 0.4400 | 0.9620 | |||
0.0248 | 0.2076 | 0.2906 | 0.9820 | 0.0048 | 0.1533 | 0.2014 | 0.9710 | −0.0150 | 0.1023 | 0.1310 | 0.9640 | ||||
0.0713 | 0.2349 | 0.3063 | 0.9820 | 0.0371 | 0.1427 | 0.2312 | 0.9710 | 0.0110 | 0.0792 | 0.1428 | 0.9690 | ||||
−0.1069 | 0.2299 | 0.6043 | 0.9830 | −0.0648 | 0.1703 | 0.45543 | 0.9800 | −0.0273 | 0.0914 | 0.2912 | 0.9770 | ||||
0.1 | 0.0342 | 0.2980 | 1.0518 | 0.9850 | 0.0365 | 0.1766 | 0.7190 | 0.9830 | 0.0244 | 0.1115 | 0.4275 | 0.9790 | |||
−0.0005 | 0.1861 | 0.3416 | 0.9920 | −0.0024 | 0.1333 | 0.1743 | 0.9830 | −0.0170 | 0.0828 | 0.0993 | 0.9770 | ||||
0.0662 | 0.2706 | 0.3296 | 0.9850 | 0.0390 | 0.1859 | 0.2283 | 0.9790 | 0.0079 | 0.1049 | 0.1405 | 0.9670 | ||||
−0.0831 | 0.2456 | 0.8136 | 0.9990 | −0.0680 | 0.1914 | 0.5084 | 0.9940 | −0.0246 | 0.1092 | 0.3080 | 0.9870 | ||||
−0.2 | −0.0708 | 0.5232 | 1.0911 | 0.9910 | −0.0421 | 0.3010 | 0.8699 | 0.9800 | −0.0405 | 0.1763 | 0.6136 | 0.9610 | |||
−0.0469 | 0.1546 | 0.5067 | 0.9940 | −0.0306 | 0.1091 | 0.2603 | 0.9740 | −0.0168 | 0.0697 | 0.1356 | 0.9630 | ||||
0.0746 | 0.5990 | 0.3029 | 0.9890 | 0.0283 | 0.2960 | 0.2443 | 0.9740 | 0.0078 | 0.1507 | 0.1721 | 0.9670 | ||||
0.0228 | 0.3499 | 1.2728 | 0.9830 | 0.0135 | 0.2384 | 0.8195 | 0.9750 | 0.0152 | 0.1336 | 0.5331 | 0.9610 | ||||
2 | 0.5 | −0.0619 | 0.1945 | 0.8590 | 0.9980 | −0.0416 | 0.1753 | 0.6607 | 0.9890 | −0.0078 | 0.1412 | 0.4426 | 0.9990 | ||
0.1125 | 0.2791 | 0.3363 | 0.9330 | 0.0729 | 0.2173 | 0.2650 | 0.9420 | 0.0159 | 0.1457 | 0.2127 | 0.9620 | ||||
0.0729 | 0.3018 | 0.3538 | 0.9800 | 0.0496 | 0.2202 | 0.2627 | 0.9720 | 0.0111 | 0.1150 | 0.1710 | 0.9670 | ||||
−0.0283 | 0.1944 | 0.9046 | 0.9700 | −0.0194 | 0.1299 | 0.7037 | 0.9890 | −0.0050 | 0.0526 | 0.4684 | 0.9620 | ||||
0.2 | 0.0729 | 0.2117 | 0.9636 | 0.9900 | 0.0547 | 0.1564 | 0.6760 | 0.9850 | 0.0360 | 0.1081 | 0.4496 | 0.9780 | |||
−0.0273 | 0.1958 | 0.3055 | 0.9910 | −0.0261 | 0.1517 | 0.2120 | 0.9720 | −0.0333 | 0.0975 | 0.1393 | 0.9610 | ||||
0.0640 | 0.4303 | 0.3059 | 0.9920 | 0.0357 | 0.3279 | 0.2164 | 0.9830 | −0.0037 | 0.1831 | 0.1390 | 0.9680 | ||||
−0.0937 | 0.2673 | 1.2401 | 0.9910 | −0.0715 | 0.2148 | 0.8714 | 0.9860 | −0.0324 | 0.1177 | 0.5893 | 0.9690 | ||||
0.1 | 0.0749 | 0.2478 | 0.9590 | 0.9990 | 0.0682 | 0.1670 | 0.6936 | 0.9880 | 0.0354 | 0.0876 | 0.4347 | 0.9720 | |||
−0.0373 | 0.1903 | 0.3065 | 0.9910 | −0.0583 | 0.1447 | 0.1788 | 0.9840 | −0.0433 | 0.0885 | 0.1002 | 0.9730 | ||||
0.0770 | 0.4571 | 0.2880 | 0.9780 | −0.0244 | 0.3578 | 0.2171 | 0.9660 | −0.0335 | 0.2197 | 0.1390 | 0.9600 | ||||
−0.1132 | 0.2756 | 1.3742 | 0.9970 | −0.0447 | 0.2279 | 0.9692 | 0.9870 | −0.0210 | 0.1420 | 0.6188 | 0.9670 |
True Value | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
bias | RMSE | SE | CP | bias | RMSE | SE | CP | bias | RMSE | SE | CP | ||||
0.5 | 1 | 0.5 | −0.0913 | 0.2344 | 1.3674 | 0.9830 | −0.0599 | 0.1952 | 1.0293 | 0.9730 | −0.0365 | 0.1549 | 0.6736 | 0.9620 | |
0.1068 | 0.2685 | 0.4348 | 0.9830 | 0.0550 | 0.2051 | 0.3611 | 0.9730 | 0.0148 | 0.1424 | 0.2791 | 0.9760 | ||||
0.0100 | 0.1186 | 0.3590 | 0.9830 | −0.0102 | 0.0794 | 0.2659 | 0.9710 | −0.0235 | 0.0514 | 0.1677 | 0.9660 | ||||
−0.0437 | 0.1403 | 0.6819 | 0.9730 | −0.0199 | 0.0769 | 0.5082 | 0.9710 | −0.0102 | 0.0192 | 0.3254 | 0.9620 | ||||
0.2 | −0.3935 | 1.6522 | 1.3146 | 0.9740 | 0.0161 | 0.1571 | 0.9409 | 0.9700 | −0.0014 | 0.0963 | 0.6257 | 0.9640 | |||
0.0066 | 0.1899 | 0.6875 | 0.9910 | −0.0069 | 0.1390 | 0.2749 | 0.9820 | −0.0142 | 0.0852 | 0.1846 | 0.9740 | ||||
0.1861 | 1.6107 | 0.2626 | 0.9910 | −0.0048 | 0.1053 | 0.1808 | 0.9720 | −0.0271 | 0.0631 | 0.1135 | 0.9640 | ||||
−0.1176 | 0.1962 | 0.9611 | 0.9940 | −0.0758 | 0.1442 | 0.5331 | 0.9800 | −0.0293 | 0.0677 | 0.3649 | 0.9740 | ||||
0.1 | 0.0042 | 0.4007 | 1.2466 | 0.9850 | 0.0154 | 0.1490 | 0.8994 | 0.9760 | 0.0025 | 0.0896 | 0.5491 | 0.9690 | |||
0.0016 | 0.1708 | 0.4576 | 0.9930 | −0.0082 | 0.1162 | 0.2241 | 0.9810 | −0.0163 | 0.0703 | 0.1307 | 0.9750 | ||||
0.0608 | 0.3558 | 0.2454 | 0.9850 | 0.0043 | 0.1201 | 0.1726 | 0.9760 | −0.0269 | 0.0726 | 0.1080 | 0.9650 | ||||
−0.1368 | 0.2118 | 0.9204 | 0.9930 | −0.0904 | 0.1640 | 0.5495 | 0.9860 | −0.0351 | 0.0812 | 0.3475 | 0.9690 | ||||
−0.2 | −0.8713 | 1.3807 | 1.3926 | 0.9870 | −0.1653 | 0.5025 | 1.0040 | 0.9720 | −0.1012 | 0.2560 | 0.6365 | 0.9610 | |||
−0.0079 | 0.1100 | 1.2154 | 0.9850 | −0.0110 | 0.0795 | 0.4796 | 0.9770 | −0.0106 | 0.0525 | 0.1965 | 0.9960 | ||||
1.6920 | 3.4927 | 0.2594 | 0.9980 | 0.1181 | 0.4368 | 0.1866 | 0.9870 | 0.0243 | 0.1559 | 0.1196 | 0.9790 | ||||
−0.1205 | 0.2113 | 1.3655 | 0.9870 | −0.0667 | 0.1424 | 1.1179 | 0.9770 | −0.0189 | 0.0593 | 0.5751 | 0.9690 | ||||
2 | 0.5 | −0.0965 | 0.2355 | 1.2621 | 0.9820 | −0.0541 | 0.1903 | 0.9674 | 0.9760 | −0.0320 | 0.1521 | 0.6883 | 0.9670 | ||
0.1020 | 0.2691 | 0.4254 | 0.9820 | 0.0378 | 0.2007 | 0.3501 | 0.9760 | 0.0082 | 0.1424 | 0.2801 | 0.9650 | ||||
−0.0087 | 0.2752 | 0.3522 | 0.9720 | −0.0554 | 0.2024 | 0.2537 | 0.9690 | −0.0565 | 0.1107 | 0.1619 | 0.9650 | ||||
−0.0279 | 0.1621 | 1.3228 | 0.9820 | −0.0034 | 0.1044 | 0.9964 | 0.9760 | −0.0076 | 0.0282 | 0.6670 | 0.9680 | ||||
0.2 | 0.0260 | 0.2239 | 1.2350 | 0.9890 | 0.0190 | 0.1622 | 0.9383 | 0.9750 | 0.1019 | 0.6279 | 0.6200 | 0.9690 | |||
−0.0117 | 0.1867 | 0.3544 | 0.9870 | −0.0280 | 0.1361 | 0.2810 | 0.9750 | −0.0241 | 0.0879 | 0.1840 | 0.9670 | ||||
−0.0061 | 0.3345 | 0.2585 | 0.9900 | −0.0570 | 0.2333 | 0.1800 | 0.9800 | −0.0743 | 0.1438 | 0.1147 | 0.9720 | ||||
−0.1002 | 0.2056 | 1.4290 | 0.9870 | −0.0636 | 0.1509 | 1.0908 | 0.9700 | −0.0263 | 0.0760 | 0.7386 | 0.9690 | ||||
0.1 | 0.0180 | 0.2640 | 1.1769 | 0.9880 | 0.0133 | 0.1712 | 0.8844 | 0.9790 | 0.0022 | 0.0750 | 0.5468 | 0.9680 | |||
−0.0212 | 0.1646 | 0.3278 | 0.9800 | −0.0311 | 0.1141 | 0.2360 | 0.9750 | −0.0263 | 0.0693 | 0.1289 | 0.9680 | ||||
0.0163 | 0.3927 | 0.2341 | 0.9830 | −0.0587 | 0.2823 | 0.1714 | 0.9780 | −0.0799 | 0.1674 | 0.1069 | 0.9680 | ||||
−0.1189 | 0.2260 | 1.4648 | 0.9830 | −0.0674 | 0.1733 | 1.1255 | 0.9790 | −0.0305 | 0.0991 | 0.6944 | 0.9680 |
Distribution | IPM-B | GP | WP | BBXII | IPM | W |
---|---|---|---|---|---|---|
−12.610 (1.5148) | 0.0037 (1.1 ) | 5.3277 (0.0043) | 15.156 (1.2348) | 0.144 (0.253) | − | |
6.0506 (0.6685) | 0.1198 (2.9 ) | 0.0280 (6.7 ) | 0.9763 (0.6613) | 2.484 (0.505) | 4.169 (0.1610) | |
0.0660 (0.0143) | 0.0997 (0.4975) | 4.2861 (0.3996) | 14.812 (0.9717) | 1.075 (0.069) | 28.565 (0.4119) | |
0.0016 (0.5050) | − | − | 0.8248 (0.5488) | − | − | |
− | − | − | 7.4191 (1.4849) | − | − | |
AIC | 1909.79 | 2174.64 | 2012.36 | 1911.41 | 1913.79 | 2063.75 |
BIC | 1924.79 | 2185.89 | 2023.61 | 1930.16 | 1926.82 | 2071.25 |
Distribution | IBM | 95% CI | Interval Length |
---|---|---|---|
IPMB | 0.1910 | (0.1836–0.1985) | 0.0149 |
BBXII | 0.1922 | (0.1847–0.1998) | 0.0151 |
IPM | 0.1938 | (0.1763–0.2113) | 0.0175 |
WP | 0.2364 | (0.2264–0.2464) | 0.0199 |
Distribution | IPM-P | GG | WG | BBXII | IPM | W |
---|---|---|---|---|---|---|
0.1398 (0.2946) | 0.0728 (0.3357) | 0.0078 (6.6 ) | 7.3236 (3.5 ) | 0.019 (0.041) | − | |
2.4751 (0.6374) | 1.6926 (0.0125) | 2.4962 (0.1401) | 11.366 (8.4 ) | 5.893 (0.309) | 2.3451 (0.2798) | |
1.0698 (0.2219) | 2.2587 (0.0225) | 0.9392 (0.0032) | 0.3442 (2.2 ) | 0.044 (0.001) | 1.3937 (0.0998) | |
0.0459 (1.9492) | − | − | 0.0673 (6.3 ) | − | − | |
− | − | − | 6.3903 (3.8 ) | − | − | |
AIC | 64.68 | 72.85 | 65.62 | 65.92 | 68.68 | 66.58 |
BIC | 71.44 | 77.92 | 71.69 | 74.36 | 76.13 | 69.96 |
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Barrios-Blanco, L.; Gallardo, D.I.; Gómez, H.J.; Bourguignon, M. A Compound Class of Inverse-Power Muth and Power Series Distributions. Axioms 2023, 12, 383. https://doi.org/10.3390/axioms12040383
Barrios-Blanco L, Gallardo DI, Gómez HJ, Bourguignon M. A Compound Class of Inverse-Power Muth and Power Series Distributions. Axioms. 2023; 12(4):383. https://doi.org/10.3390/axioms12040383
Chicago/Turabian StyleBarrios-Blanco, Leonardo, Diego I. Gallardo, Héctor J. Gómez, and Marcelo Bourguignon. 2023. "A Compound Class of Inverse-Power Muth and Power Series Distributions" Axioms 12, no. 4: 383. https://doi.org/10.3390/axioms12040383