R Under development (unstable) (2024-10-15 r87238 ucrt) -- "Unsuffered Consequences" Copyright (C) 2024 The R Foundation for Statistical Computing Platform: x86_64-w64-mingw32/x64 R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > library(RTDE) Loading required package: parallel > > # ?FGM > > ##### > # (1) density function > u <- v <- seq(0, 1, length=25) > > cbind(u, v, dFGM(u, v, 1/2)) u v [1,] 0.00000000 0.00000000 1.500000 [2,] 0.04166667 0.04166667 1.420139 [3,] 0.08333333 0.08333333 1.347222 [4,] 0.12500000 0.12500000 1.281250 [5,] 0.16666667 0.16666667 1.222222 [6,] 0.20833333 0.20833333 1.170139 [7,] 0.25000000 0.25000000 1.125000 [8,] 0.29166667 0.29166667 1.086806 [9,] 0.33333333 0.33333333 1.055556 [10,] 0.37500000 0.37500000 1.031250 [11,] 0.41666667 0.41666667 1.013889 [12,] 0.45833333 0.45833333 1.003472 [13,] 0.50000000 0.50000000 1.000000 [14,] 0.54166667 0.54166667 1.003472 [15,] 0.58333333 0.58333333 1.013889 [16,] 0.62500000 0.62500000 1.031250 [17,] 0.66666667 0.66666667 1.055556 [18,] 0.70833333 0.70833333 1.086806 [19,] 0.75000000 0.75000000 1.125000 [20,] 0.79166667 0.79166667 1.170139 [21,] 0.83333333 0.83333333 1.222222 [22,] 0.87500000 0.87500000 1.281250 [23,] 0.91666667 0.91666667 1.347222 [24,] 0.95833333 0.95833333 1.420139 [25,] 1.00000000 1.00000000 1.500000 > cbind(u, v, outer(u, v, dFGM, alpha=1/2)) u v [1,] 0.00000000 0.00000000 1.5000000 1.4583333 1.4166667 1.37500 1.3333333 [2,] 0.04166667 0.04166667 1.4583333 1.4201389 1.3819444 1.34375 1.3055556 [3,] 0.08333333 0.08333333 1.4166667 1.3819444 1.3472222 1.31250 1.2777778 [4,] 0.12500000 0.12500000 1.3750000 1.3437500 1.3125000 1.28125 1.2500000 [5,] 0.16666667 0.16666667 1.3333333 1.3055556 1.2777778 1.25000 1.2222222 [6,] 0.20833333 0.20833333 1.2916667 1.2673611 1.2430556 1.21875 1.1944444 [7,] 0.25000000 0.25000000 1.2500000 1.2291667 1.2083333 1.18750 1.1666667 [8,] 0.29166667 0.29166667 1.2083333 1.1909722 1.1736111 1.15625 1.1388889 [9,] 0.33333333 0.33333333 1.1666667 1.1527778 1.1388889 1.12500 1.1111111 [10,] 0.37500000 0.37500000 1.1250000 1.1145833 1.1041667 1.09375 1.0833333 [11,] 0.41666667 0.41666667 1.0833333 1.0763889 1.0694444 1.06250 1.0555556 [12,] 0.45833333 0.45833333 1.0416667 1.0381944 1.0347222 1.03125 1.0277778 [13,] 0.50000000 0.50000000 1.0000000 1.0000000 1.0000000 1.00000 1.0000000 [14,] 0.54166667 0.54166667 0.9583333 0.9618056 0.9652778 0.96875 0.9722222 [15,] 0.58333333 0.58333333 0.9166667 0.9236111 0.9305556 0.93750 0.9444444 [16,] 0.62500000 0.62500000 0.8750000 0.8854167 0.8958333 0.90625 0.9166667 [17,] 0.66666667 0.66666667 0.8333333 0.8472222 0.8611111 0.87500 0.8888889 [18,] 0.70833333 0.70833333 0.7916667 0.8090278 0.8263889 0.84375 0.8611111 [19,] 0.75000000 0.75000000 0.7500000 0.7708333 0.7916667 0.81250 0.8333333 [20,] 0.79166667 0.79166667 0.7083333 0.7326389 0.7569444 0.78125 0.8055556 [21,] 0.83333333 0.83333333 0.6666667 0.6944444 0.7222222 0.75000 0.7777778 [22,] 0.87500000 0.87500000 0.6250000 0.6562500 0.6875000 0.71875 0.7500000 [23,] 0.91666667 0.91666667 0.5833333 0.6180556 0.6527778 0.68750 0.7222222 [24,] 0.95833333 0.95833333 0.5416667 0.5798611 0.6180556 0.65625 0.6944444 [25,] 1.00000000 1.00000000 0.5000000 0.5416667 0.5833333 0.62500 0.6666667 [1,] 1.2916667 1.2500000 1.2083333 1.1666667 1.1250000 1.0833333 1.0416667 1 [2,] 1.2673611 1.2291667 1.1909722 1.1527778 1.1145833 1.0763889 1.0381944 1 [3,] 1.2430556 1.2083333 1.1736111 1.1388889 1.1041667 1.0694444 1.0347222 1 [4,] 1.2187500 1.1875000 1.1562500 1.1250000 1.0937500 1.0625000 1.0312500 1 [5,] 1.1944444 1.1666667 1.1388889 1.1111111 1.0833333 1.0555556 1.0277778 1 [6,] 1.1701389 1.1458333 1.1215278 1.0972222 1.0729167 1.0486111 1.0243056 1 [7,] 1.1458333 1.1250000 1.1041667 1.0833333 1.0625000 1.0416667 1.0208333 1 [8,] 1.1215278 1.1041667 1.0868056 1.0694444 1.0520833 1.0347222 1.0173611 1 [9,] 1.0972222 1.0833333 1.0694444 1.0555556 1.0416667 1.0277778 1.0138889 1 [10,] 1.0729167 1.0625000 1.0520833 1.0416667 1.0312500 1.0208333 1.0104167 1 [11,] 1.0486111 1.0416667 1.0347222 1.0277778 1.0208333 1.0138889 1.0069444 1 [12,] 1.0243056 1.0208333 1.0173611 1.0138889 1.0104167 1.0069444 1.0034722 1 [13,] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1 [14,] 0.9756944 0.9791667 0.9826389 0.9861111 0.9895833 0.9930556 0.9965278 1 [15,] 0.9513889 0.9583333 0.9652778 0.9722222 0.9791667 0.9861111 0.9930556 1 [16,] 0.9270833 0.9375000 0.9479167 0.9583333 0.9687500 0.9791667 0.9895833 1 [17,] 0.9027778 0.9166667 0.9305556 0.9444444 0.9583333 0.9722222 0.9861111 1 [18,] 0.8784722 0.8958333 0.9131944 0.9305556 0.9479167 0.9652778 0.9826389 1 [19,] 0.8541667 0.8750000 0.8958333 0.9166667 0.9375000 0.9583333 0.9791667 1 [20,] 0.8298611 0.8541667 0.8784722 0.9027778 0.9270833 0.9513889 0.9756944 1 [21,] 0.8055556 0.8333333 0.8611111 0.8888889 0.9166667 0.9444444 0.9722222 1 [22,] 0.7812500 0.8125000 0.8437500 0.8750000 0.9062500 0.9375000 0.9687500 1 [23,] 0.7569444 0.7916667 0.8263889 0.8611111 0.8958333 0.9305556 0.9652778 1 [24,] 0.7326389 0.7708333 0.8090278 0.8472222 0.8854167 0.9236111 0.9618056 1 [25,] 0.7083333 0.7500000 0.7916667 0.8333333 0.8750000 0.9166667 0.9583333 1 [1,] 0.9583333 0.9166667 0.8750000 0.8333333 0.7916667 0.7500000 0.7083333 [2,] 0.9618056 0.9236111 0.8854167 0.8472222 0.8090278 0.7708333 0.7326389 [3,] 0.9652778 0.9305556 0.8958333 0.8611111 0.8263889 0.7916667 0.7569444 [4,] 0.9687500 0.9375000 0.9062500 0.8750000 0.8437500 0.8125000 0.7812500 [5,] 0.9722222 0.9444444 0.9166667 0.8888889 0.8611111 0.8333333 0.8055556 [6,] 0.9756944 0.9513889 0.9270833 0.9027778 0.8784722 0.8541667 0.8298611 [7,] 0.9791667 0.9583333 0.9375000 0.9166667 0.8958333 0.8750000 0.8541667 [8,] 0.9826389 0.9652778 0.9479167 0.9305556 0.9131944 0.8958333 0.8784722 [9,] 0.9861111 0.9722222 0.9583333 0.9444444 0.9305556 0.9166667 0.9027778 [10,] 0.9895833 0.9791667 0.9687500 0.9583333 0.9479167 0.9375000 0.9270833 [11,] 0.9930556 0.9861111 0.9791667 0.9722222 0.9652778 0.9583333 0.9513889 [12,] 0.9965278 0.9930556 0.9895833 0.9861111 0.9826389 0.9791667 0.9756944 [13,] 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 [14,] 1.0034722 1.0069444 1.0104167 1.0138889 1.0173611 1.0208333 1.0243056 [15,] 1.0069444 1.0138889 1.0208333 1.0277778 1.0347222 1.0416667 1.0486111 [16,] 1.0104167 1.0208333 1.0312500 1.0416667 1.0520833 1.0625000 1.0729167 [17,] 1.0138889 1.0277778 1.0416667 1.0555556 1.0694444 1.0833333 1.0972222 [18,] 1.0173611 1.0347222 1.0520833 1.0694444 1.0868056 1.1041667 1.1215278 [19,] 1.0208333 1.0416667 1.0625000 1.0833333 1.1041667 1.1250000 1.1458333 [20,] 1.0243056 1.0486111 1.0729167 1.0972222 1.1215278 1.1458333 1.1701389 [21,] 1.0277778 1.0555556 1.0833333 1.1111111 1.1388889 1.1666667 1.1944444 [22,] 1.0312500 1.0625000 1.0937500 1.1250000 1.1562500 1.1875000 1.2187500 [23,] 1.0347222 1.0694444 1.1041667 1.1388889 1.1736111 1.2083333 1.2430556 [24,] 1.0381944 1.0763889 1.1145833 1.1527778 1.1909722 1.2291667 1.2673611 [25,] 1.0416667 1.0833333 1.1250000 1.1666667 1.2083333 1.2500000 1.2916667 [1,] 0.6666667 0.62500 0.5833333 0.5416667 0.5000000 [2,] 0.6944444 0.65625 0.6180556 0.5798611 0.5416667 [3,] 0.7222222 0.68750 0.6527778 0.6180556 0.5833333 [4,] 0.7500000 0.71875 0.6875000 0.6562500 0.6250000 [5,] 0.7777778 0.75000 0.7222222 0.6944444 0.6666667 [6,] 0.8055556 0.78125 0.7569444 0.7326389 0.7083333 [7,] 0.8333333 0.81250 0.7916667 0.7708333 0.7500000 [8,] 0.8611111 0.84375 0.8263889 0.8090278 0.7916667 [9,] 0.8888889 0.87500 0.8611111 0.8472222 0.8333333 [10,] 0.9166667 0.90625 0.8958333 0.8854167 0.8750000 [11,] 0.9444444 0.93750 0.9305556 0.9236111 0.9166667 [12,] 0.9722222 0.96875 0.9652778 0.9618056 0.9583333 [13,] 1.0000000 1.00000 1.0000000 1.0000000 1.0000000 [14,] 1.0277778 1.03125 1.0347222 1.0381944 1.0416667 [15,] 1.0555556 1.06250 1.0694444 1.0763889 1.0833333 [16,] 1.0833333 1.09375 1.1041667 1.1145833 1.1250000 [17,] 1.1111111 1.12500 1.1388889 1.1527778 1.1666667 [18,] 1.1388889 1.15625 1.1736111 1.1909722 1.2083333 [19,] 1.1666667 1.18750 1.2083333 1.2291667 1.2500000 [20,] 1.1944444 1.21875 1.2430556 1.2673611 1.2916667 [21,] 1.2222222 1.25000 1.2777778 1.3055556 1.3333333 [22,] 1.2500000 1.28125 1.3125000 1.3437500 1.3750000 [23,] 1.2777778 1.31250 1.3472222 1.3819444 1.4166667 [24,] 1.3055556 1.34375 1.3819444 1.4201389 1.4583333 [25,] 1.3333333 1.37500 1.4166667 1.4583333 1.5000000 > > > ##### > # (2) distribution function > > cbind(u, v, pFGM(u, v, 1/2)) u v [1,] 0.00000000 0.00000000 0.000000000 [2,] 0.04166667 0.04166667 0.002533336 [3,] 0.08333333 0.08333333 0.009862076 [4,] 0.12500000 0.12500000 0.021606445 [5,] 0.16666667 0.16666667 0.037422840 [6,] 0.20833333 0.20833333 0.057003822 [7,] 0.25000000 0.25000000 0.080078125 [8,] 0.29166667 0.29166667 0.106410651 [9,] 0.33333333 0.33333333 0.135802469 [10,] 0.37500000 0.37500000 0.168090820 [11,] 0.41666667 0.41666667 0.203149113 [12,] 0.45833333 0.45833333 0.240886924 [13,] 0.50000000 0.50000000 0.281250000 [14,] 0.54166667 0.54166667 0.324220257 [15,] 0.58333333 0.58333333 0.369815779 [16,] 0.62500000 0.62500000 0.418090820 [17,] 0.66666667 0.66666667 0.469135802 [18,] 0.70833333 0.70833333 0.523077317 [19,] 0.75000000 0.75000000 0.580078125 [20,] 0.79166667 0.79166667 0.640337155 [21,] 0.83333333 0.83333333 0.704089506 [22,] 0.87500000 0.87500000 0.771606445 [23,] 0.91666667 0.91666667 0.843195409 [24,] 0.95833333 0.95833333 0.919200002 [25,] 1.00000000 1.00000000 1.000000000 > cbind(u, v, outer(u, v, pFGM, alpha=1/2)) u v [1,] 0.00000000 0.00000000 0 0.000000000 0.000000000 0.000000000 0.00000000 [2,] 0.04166667 0.04166667 0 0.002533336 0.004997348 0.007392036 0.00971740 [3,] 0.08333333 0.08333333 0 0.004997348 0.009862076 0.014594184 0.01919367 [4,] 0.12500000 0.12500000 0 0.007392036 0.014594184 0.021606445 0.02842882 [5,] 0.16666667 0.16666667 0 0.009717400 0.019193673 0.028428819 0.03742284 [6,] 0.20833333 0.20833333 0 0.011973440 0.023660542 0.035061306 0.04617573 [7,] 0.25000000 0.25000000 0 0.014160156 0.027994792 0.041503906 0.05468750 [8,] 0.29166667 0.29166667 0 0.016277549 0.032196422 0.047756619 0.06295814 [9,] 0.33333333 0.33333333 0 0.018325617 0.036265432 0.053819444 0.07098765 [10,] 0.37500000 0.37500000 0 0.020304362 0.040201823 0.059692383 0.07877604 [11,] 0.41666667 0.41666667 0 0.022213783 0.044005594 0.065375434 0.08632330 [12,] 0.45833333 0.45833333 0 0.024053880 0.047676746 0.070868598 0.09362944 [13,] 0.50000000 0.50000000 0 0.025824653 0.051215278 0.076171875 0.10069444 [14,] 0.54166667 0.54166667 0 0.027526102 0.054621190 0.081285265 0.10751833 [15,] 0.58333333 0.58333333 0 0.029158227 0.057894483 0.086208767 0.11410108 [16,] 0.62500000 0.62500000 0 0.030721029 0.061035156 0.090942383 0.12044271 [17,] 0.66666667 0.66666667 0 0.032214506 0.064043210 0.095486111 0.12654321 [18,] 0.70833333 0.70833333 0 0.033638660 0.066918644 0.099839952 0.13240258 [19,] 0.75000000 0.75000000 0 0.034993490 0.069661458 0.104003906 0.13802083 [20,] 0.79166667 0.79166667 0 0.036278995 0.072271653 0.107977973 0.14339796 [21,] 0.83333333 0.83333333 0 0.037495177 0.074749228 0.111762153 0.14853395 [22,] 0.87500000 0.87500000 0 0.038642036 0.077094184 0.115356445 0.15342882 [23,] 0.91666667 0.91666667 0 0.039719570 0.079306520 0.118760851 0.15808256 [24,] 0.95833333 0.95833333 0 0.040727780 0.081386236 0.121975369 0.16249518 [25,] 1.00000000 1.00000000 0 0.041666667 0.083333333 0.125000000 0.16666667 [1,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 [2,] 0.01197344 0.01416016 0.01627755 0.01832562 0.02030436 0.02221378 [3,] 0.02366054 0.02799479 0.03219642 0.03626543 0.04020182 0.04400559 [4,] 0.03506131 0.04150391 0.04775662 0.05381944 0.05969238 0.06537543 [5,] 0.04617573 0.05468750 0.06295814 0.07098765 0.07877604 0.08632330 [6,] 0.05700382 0.06754557 0.07780099 0.08777006 0.09745280 0.10684920 [7,] 0.06754557 0.08007812 0.09228516 0.10416667 0.11572266 0.12695312 [8,] 0.07780099 0.09228516 0.10641065 0.12017747 0.13358561 0.14663508 [9,] 0.08777006 0.10416667 0.12017747 0.13580247 0.15104167 0.16589506 [10,] 0.09745280 0.11572266 0.13358561 0.15104167 0.16809082 0.18473307 [11,] 0.10684920 0.12695312 0.14663508 0.16589506 0.18473307 0.20314911 [12,] 0.11595926 0.13785807 0.15932587 0.18036265 0.20096842 0.22114318 [13,] 0.12478299 0.14843750 0.17165799 0.19444444 0.21679688 0.23871528 [14,] 0.13332037 0.15869141 0.18363143 0.20814043 0.23221842 0.25586540 [15,] 0.14157142 0.16861979 0.19524619 0.22145062 0.24723307 0.27259356 [16,] 0.14953613 0.17822266 0.20650228 0.23437500 0.26184082 0.28889974 [17,] 0.15721451 0.18750000 0.21739969 0.24691358 0.27604167 0.30478395 [18,] 0.16460654 0.19645182 0.22793843 0.25906636 0.28983561 0.32024619 [19,] 0.17171224 0.20507812 0.23811849 0.27083333 0.30322266 0.33528646 [20,] 0.17853160 0.21337891 0.24793988 0.28221451 0.31620280 0.34990476 [21,] 0.18506462 0.22135417 0.25740258 0.29320988 0.32877604 0.36410108 [22,] 0.19131131 0.22900391 0.26650662 0.30381944 0.34094238 0.37787543 [23,] 0.19727165 0.23632812 0.27525198 0.31404321 0.35270182 0.39122782 [24,] 0.20294566 0.24332682 0.28363866 0.32388117 0.36405436 0.40415823 [25,] 0.20833333 0.25000000 0.29166667 0.33333333 0.37500000 0.41666667 [1,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 [2,] 0.02405388 0.02582465 0.02752610 0.02915823 0.03072103 0.03221451 [3,] 0.04767675 0.05121528 0.05462119 0.05789448 0.06103516 0.06404321 [4,] 0.07086860 0.07617188 0.08128526 0.08620877 0.09094238 0.09548611 [5,] 0.09362944 0.10069444 0.10751833 0.11410108 0.12044271 0.12654321 [6,] 0.11595926 0.12478299 0.13332037 0.14157142 0.14953613 0.15721451 [7,] 0.13785807 0.14843750 0.15869141 0.16861979 0.17822266 0.18750000 [8,] 0.15932587 0.17165799 0.18363143 0.19524619 0.20650228 0.21739969 [9,] 0.18036265 0.19444444 0.20814043 0.22145062 0.23437500 0.24691358 [10,] 0.20096842 0.21679688 0.23221842 0.24723307 0.26184082 0.27604167 [11,] 0.22114318 0.23871528 0.25586540 0.27259356 0.28889974 0.30478395 [12,] 0.24088692 0.26019965 0.27908137 0.29753207 0.31555176 0.33314043 [13,] 0.26019965 0.28125000 0.30186632 0.32204861 0.34179688 0.36111111 [14,] 0.27908137 0.30186632 0.32422026 0.34614318 0.36763509 0.38869599 [15,] 0.29753207 0.32204861 0.34614318 0.36981578 0.39306641 0.41589506 [16,] 0.31555176 0.34179688 0.36763509 0.39306641 0.41809082 0.44270833 [17,] 0.33314043 0.36111111 0.38869599 0.41589506 0.44270833 0.46913580 [18,] 0.35029809 0.37999132 0.40932587 0.43830175 0.46691895 0.49517747 [19,] 0.36702474 0.39843750 0.42952474 0.46028646 0.49072266 0.52083333 [20,] 0.38332037 0.41644965 0.44929260 0.48184920 0.51411947 0.54610340 [21,] 0.39918499 0.43402778 0.46862944 0.50298997 0.53710937 0.57098765 [22,] 0.41461860 0.45117188 0.48753526 0.52370877 0.55969238 0.59548611 [23,] 0.42962119 0.46788194 0.50601008 0.54400559 0.58186849 0.61959877 [24,] 0.44419277 0.48415799 0.52405388 0.56388045 0.60363770 0.64332562 [25,] 0.45833333 0.50000000 0.54166667 0.58333333 0.62500000 0.66666667 [1,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 [2,] 0.03363866 0.03499349 0.03627900 0.03749518 0.03864204 0.03971957 [3,] 0.06691864 0.06966146 0.07227165 0.07474923 0.07709418 0.07930652 [4,] 0.09983995 0.10400391 0.10797797 0.11176215 0.11535645 0.11876085 [5,] 0.13240258 0.13802083 0.14339796 0.14853395 0.15342882 0.15808256 [6,] 0.16460654 0.17171224 0.17853160 0.18506462 0.19131131 0.19727165 [7,] 0.19645182 0.20507812 0.21337891 0.22135417 0.22900391 0.23632812 [8,] 0.22793843 0.23811849 0.24793988 0.25740258 0.26650662 0.27525198 [9,] 0.25906636 0.27083333 0.28221451 0.29320988 0.30381944 0.31404321 [10,] 0.28983561 0.30322266 0.31620280 0.32877604 0.34094238 0.35270182 [11,] 0.32024619 0.33528646 0.34990476 0.36410108 0.37787543 0.39122782 [12,] 0.35029809 0.36702474 0.38332037 0.39918499 0.41461860 0.42962119 [13,] 0.37999132 0.39843750 0.41644965 0.43402778 0.45117188 0.46788194 [14,] 0.40932587 0.42952474 0.44929260 0.46862944 0.48753526 0.50601008 [15,] 0.43830175 0.46028646 0.48184920 0.50298997 0.52370877 0.54400559 [16,] 0.46691895 0.49072266 0.51411947 0.53710937 0.55969238 0.58186849 [17,] 0.49517747 0.52083333 0.54610340 0.57098765 0.59548611 0.61959877 [18,] 0.52307732 0.55061849 0.57780099 0.60462481 0.63108995 0.65719642 [19,] 0.55061849 0.58007812 0.60921224 0.63802083 0.66650391 0.69466146 [20,] 0.57780099 0.60921224 0.64033716 0.67117573 0.70172797 0.73199388 [21,] 0.60462481 0.63802083 0.67117573 0.70408951 0.73676215 0.76919367 [22,] 0.63108995 0.66650391 0.70172797 0.73676215 0.77160645 0.80626085 [23,] 0.65719642 0.69466146 0.73199388 0.76919367 0.80626085 0.84319541 [24,] 0.68294422 0.72249349 0.76197344 0.80138407 0.84072537 0.87999735 [25,] 0.70833333 0.75000000 0.79166667 0.83333333 0.87500000 0.91666667 [1,] 0.00000000 0.00000000 [2,] 0.04072778 0.04166667 [3,] 0.08138624 0.08333333 [4,] 0.12197537 0.12500000 [5,] 0.16249518 0.16666667 [6,] 0.20294566 0.20833333 [7,] 0.24332682 0.25000000 [8,] 0.28363866 0.29166667 [9,] 0.32388117 0.33333333 [10,] 0.36405436 0.37500000 [11,] 0.40415823 0.41666667 [12,] 0.44419277 0.45833333 [13,] 0.48415799 0.50000000 [14,] 0.52405388 0.54166667 [15,] 0.56388045 0.58333333 [16,] 0.60363770 0.62500000 [17,] 0.64332562 0.66666667 [18,] 0.68294422 0.70833333 [19,] 0.72249349 0.75000000 [20,] 0.76197344 0.79166667 [21,] 0.80138407 0.83333333 [22,] 0.84072537 0.87500000 [23,] 0.87999735 0.91666667 [24,] 0.91920000 0.95833333 [25,] 0.95833333 1.00000000 > > > > ##### > # (3) survival probabilities > > checkFGMfrechet <- function(x, omegatilde, beta) + { + v <- 1-exp(-1/(x*omegatilde)) + u <- 1-exp(-1/x) + u+v-1+(1-u)*(1-v)*(1+beta*u*v) + } > > > x <- 1:20 > p <- pfrechet(x, 1, 0) > pFGM(p, p, 1/2, lower.tail=FALSE) [1] 0.426614794 0.183295324 0.100982185 0.063767591 0.043871409 0.032011416 [7] 0.024380144 0.019183421 0.015486354 0.012763096 0.010699522 0.009098630 [13] 0.007831804 0.006812190 0.005979425 0.005290502 0.004714109 0.004227014 [19] 0.003811681 0.003454678 > checkFGMfrechet(x, 1, 1/2) [1] 0.426614794 0.183295324 0.100982185 0.063767591 0.043871409 0.032011416 [7] 0.024380144 0.019183421 0.015486354 0.012763096 0.010699522 0.009098630 [13] 0.007831804 0.006812190 0.005979425 0.005290502 0.004714109 0.004227014 [19] 0.003811681 0.003454678 > > > y <- 1:20 > p2 <- pfrechet(2*y, 1, 0) > pFGM(p, p2, 1/2, lower.tail=FALSE) [1] 0.276468533 0.107591347 0.056715011 0.034923465 0.023639625 0.017054395 [7] 0.012880555 0.010070145 0.008088191 0.006638464 0.005546186 0.004702838 [13] 0.004038158 0.003505028 0.003070896 0.002712689 0.002413683 0.002161517 [19] 0.001946896 0.001762721 > checkFGMfrechet(x, 2, 1/2) [1] 0.276468533 0.107591347 0.056715011 0.034923465 0.023639625 0.017054395 [7] 0.012880555 0.010070145 0.008088191 0.006638464 0.005546186 0.004702838 [13] 0.004038158 0.003505028 0.003070896 0.002712689 0.002413683 0.002161517 [19] 0.001946896 0.001762721 > > > ##### > # (4) simulation > > n <- 1e6 > > uv <- rFGM(n, 1/2) > S <- function(x, y) sum(uv[,1] > x & uv[,2] > y) / NROW(uv) > > sapply(1:9/10, function(z) + c(S(z, 1/4), + pFGM(z, 1/4, 1/2, lower.tail=FALSE))) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 0.6840170 0.615716 0.5451650 0.472939 0.3988850 0.323345 0.2451120 [2,] 0.6834375 0.615000 0.5446875 0.472500 0.3984375 0.322500 0.2446875 [,8] [,9] [1,] 0.165605 0.0839550 [2,] 0.165000 0.0834375 > > sapply(1:9/10, function(z) + c(S(z, 1/2), + pFGM(z, 1/2, 1/2, lower.tail=FALSE))) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [1,] 0.46173 0.42081 0.376794 0.330551 0.281681 0.230618 0.176441 0.120413 [2,] 0.46125 0.42000 0.376250 0.330000 0.281250 0.230000 0.176250 0.120000 [,9] [1,] 0.061625 [2,] 0.061250 > > sapply(1:9/10, function(z) + c(S(z, 3/4), + pFGM(z, 3/4, 1/2, lower.tail=FALSE))) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [1,] 0.2336640 0.215363 0.1948890 0.172934 0.1487350 0.122847 0.0947730 [2,] 0.2334375 0.215000 0.1946875 0.172500 0.1484375 0.122500 0.0946875 [,8] [,9] [1,] 0.065057 0.0334170 [2,] 0.065000 0.0334375 > > > xy <- qufrechet(uv) > S <- function(x, y) sum(xy[,1] > x & xy[,2] > y) / NROW(xy) > > res <- sapply(1:9*10, function(z) + c(S(z, 10), + pFGM(pufrechet(z), pufrechet(10), 1/2, lower.tail=FALSE))) > > res <- sapply(1:9*10, function(z) + c(S(z, 40), + pFGM(pufrechet(z), pufrechet(40), 1/2, lower.tail=FALSE))) > > > plot(1:9*10, res[1,], type="l", ylim=range(res)) > lines(1:9*10, res[2,], col="red", lty=2) > > proc.time() user system elapsed 2.00 1.42 3.40