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, dfrank(u, v, 1/2)) u v [1,] 0.00000000 0.00000000 1.2707470 [2,] 0.04166667 0.04166667 1.2202054 [3,] 0.08333333 0.08333333 1.1741136 [4,] 0.12500000 0.12500000 1.1319910 [5,] 0.16666667 0.16666667 1.0934297 [6,] 0.20833333 0.20833333 1.0580817 [7,] 0.25000000 0.25000000 1.0256492 [8,] 0.29166667 0.29166667 0.9958764 [9,] 0.33333333 0.33333333 0.9685435 [10,] 0.37500000 0.37500000 0.9434610 [11,] 0.41666667 0.41666667 0.9204664 [12,] 0.45833333 0.45833333 0.8994201 [13,] 0.50000000 0.50000000 0.8802029 [14,] 0.54166667 0.54166667 0.8627142 [15,] 0.58333333 0.58333333 0.8468700 [16,] 0.62500000 0.62500000 0.8326014 [17,] 0.66666667 0.66666667 0.8198543 [18,] 0.70833333 0.70833333 0.8085883 [19,] 0.75000000 0.75000000 0.7987764 [20,] 0.79166667 0.79166667 0.7904055 [21,] 0.83333333 0.83333333 0.7834766 [22,] 0.87500000 0.87500000 0.7780053 [23,] 0.91666667 0.91666667 0.7740234 [24,] 0.95833333 0.95833333 0.7715806 [25,] 1.00000000 1.00000000 0.7707470 > cbind(u, v, outer(u, v, dfrank, alpha=1/2)) u v [1,] 0.00000000 0.00000000 1.2707470 1.2445470 1.2188872 1.1937564 1.1691437 [2,] 0.04166667 0.04166667 1.2445470 1.2202054 1.1963147 1.1728673 1.1498561 [3,] 0.08333333 0.08333333 1.2188872 1.1963147 1.1741136 1.1522795 1.1308081 [4,] 0.12500000 0.12500000 1.1937564 1.1728673 1.1522795 1.1319910 1.1119994 [5,] 0.16666667 0.16666667 1.1691437 1.1498561 1.1308081 1.1119994 1.0934297 [6,] 0.20833333 0.20833333 1.1450385 1.1272737 1.1096949 1.0923027 1.0750981 [7,] 0.25000000 0.25000000 1.1214303 1.1051133 1.0889354 1.0728986 1.0570042 [8,] 0.29166667 0.29166667 1.0983089 1.0833675 1.0685254 1.0537845 1.0391472 [9,] 0.33333333 0.33333333 1.0756641 1.0620297 1.0484602 1.0349583 1.0215263 [10,] 0.37500000 0.37500000 1.0534863 1.0410927 1.0287356 1.0164173 1.0041405 [11,] 0.41666667 0.41666667 1.0317657 1.0205500 1.0093470 0.9981592 0.9869888 [12,] 0.45833333 0.45833333 1.0104930 1.0003948 0.9902901 0.9801813 0.9700703 [13,] 0.50000000 0.50000000 0.9896588 0.9806204 0.9715605 0.9624810 0.9533838 [14,] 0.54166667 0.54166667 0.9692542 0.9612204 0.9531538 0.9450558 0.9369282 [15,] 0.58333333 0.58333333 0.9492703 0.9421883 0.9350655 0.9279030 0.9207020 [16,] 0.62500000 0.62500000 0.9296984 0.9235178 0.9172913 0.9110198 0.9047041 [17,] 0.66666667 0.66666667 0.9105300 0.9052026 0.8998269 0.8944035 0.8889330 [18,] 0.70833333 0.70833333 0.8917569 0.8872365 0.8826679 0.8780515 0.8733874 [19,] 0.75000000 0.75000000 0.8733708 0.8696134 0.8658100 0.8619608 0.8580656 [20,] 0.79166667 0.79166667 0.8553638 0.8523272 0.8492489 0.8461286 0.8429662 [21,] 0.83333333 0.83333333 0.8377281 0.8353722 0.8329804 0.8305523 0.8280876 [22,] 0.87500000 0.87500000 0.8204560 0.8187424 0.8170001 0.8152288 0.8134281 [23,] 0.91666667 0.91666667 0.8035399 0.8024320 0.8013039 0.8001553 0.7989860 [24,] 0.95833333 0.95833333 0.7869727 0.7864354 0.7858876 0.7853291 0.7847596 [25,] 1.00000000 1.00000000 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 [1,] 1.1450385 1.1214303 1.0983089 1.0756641 1.0534863 1.0317657 1.0104930 [2,] 1.1272737 1.1051133 1.0833675 1.0620297 1.0410927 1.0205500 1.0003948 [3,] 1.1096949 1.0889354 1.0685254 1.0484602 1.0287356 1.0093470 0.9902901 [4,] 1.0923027 1.0728986 1.0537845 1.0349583 1.0164173 0.9981592 0.9801813 [5,] 1.0750981 1.0570042 1.0391472 1.0215263 1.0041405 0.9869888 0.9700703 [6,] 1.0580817 1.0412540 1.0246154 1.0081665 0.9919074 0.9758384 0.9599595 [7,] 1.0412540 1.0256492 1.0101912 0.9948813 0.9797206 0.9647102 0.9498510 [8,] 1.0246154 1.0101912 0.9958764 0.9816729 0.9675824 0.9536066 0.9397471 [9,] 1.0081665 0.9948813 0.9816729 0.9685435 0.9554951 0.9425299 0.9296498 [10,] 0.9919074 0.9797206 0.9675824 0.9554951 0.9434610 0.9314824 0.9195615 [11,] 0.9758384 0.9647102 0.9536066 0.9425299 0.9314824 0.9204664 0.9094841 [12,] 0.9599595 0.9498510 0.9397471 0.9296498 0.9195615 0.9094841 0.8994201 [13,] 0.9442709 0.9351441 0.9260054 0.9168568 0.9077003 0.8985378 0.8893714 [14,] 0.9287724 0.9205901 0.9123830 0.9041528 0.8959011 0.8876296 0.8793402 [15,] 0.9134639 0.9061898 0.8988813 0.8915394 0.8841658 0.8767617 0.8693286 [16,] 0.8983452 0.8919440 0.8855014 0.8790186 0.8724965 0.8659362 0.8593389 [17,] 0.8834160 0.8778530 0.8722448 0.8665919 0.8608952 0.8551552 0.8493729 [18,] 0.8686760 0.8639175 0.8591124 0.8542610 0.8493636 0.8444208 0.8394329 [19,] 0.8541247 0.8501379 0.8461054 0.8420274 0.8379038 0.8337349 0.8295208 [20,] 0.8397616 0.8365145 0.8332248 0.8298925 0.8265175 0.8230996 0.8196387 [21,] 0.8255861 0.8230475 0.8204715 0.8178579 0.8152063 0.8125167 0.8097886 [22,] 0.8115977 0.8097372 0.8078463 0.8059248 0.8039721 0.8019882 0.7999725 [23,] 0.7977955 0.7965837 0.7953501 0.7940945 0.7928165 0.7915158 0.7901922 [24,] 0.7841789 0.7835869 0.7829834 0.7823682 0.7817409 0.7811016 0.7804498 [25,] 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 [1,] 0.9896588 0.9692542 0.9492703 0.9296984 0.9105300 0.8917569 0.8733708 [2,] 0.9806204 0.9612204 0.9421883 0.9235178 0.9052026 0.8872365 0.8696134 [3,] 0.9715605 0.9531538 0.9350655 0.9172913 0.8998269 0.8826679 0.8658100 [4,] 0.9624810 0.9450558 0.9279030 0.9110198 0.8944035 0.8780515 0.8619608 [5,] 0.9533838 0.9369282 0.9207020 0.9047041 0.8889330 0.8733874 0.8580656 [6,] 0.9442709 0.9287724 0.9134639 0.8983452 0.8834160 0.8686760 0.8541247 [7,] 0.9351441 0.9205901 0.9061898 0.8919440 0.8778530 0.8639175 0.8501379 [8,] 0.9260054 0.9123830 0.8988813 0.8855014 0.8722448 0.8591124 0.8461054 [9,] 0.9168568 0.9041528 0.8915394 0.8790186 0.8665919 0.8542610 0.8420274 [10,] 0.9077003 0.8959011 0.8841658 0.8724965 0.8608952 0.8493636 0.8379038 [11,] 0.8985378 0.8876296 0.8767617 0.8659362 0.8551552 0.8444208 0.8337349 [12,] 0.8893714 0.8793402 0.8693286 0.8593389 0.8493729 0.8394329 0.8295208 [13,] 0.8802029 0.8710345 0.8618680 0.8527055 0.8435490 0.8344004 0.8252618 [14,] 0.8710345 0.8627142 0.8543813 0.8460374 0.8376843 0.8293239 0.8209579 [15,] 0.8618680 0.8543813 0.8468700 0.8393356 0.8317797 0.8242038 0.8166095 [16,] 0.8527055 0.8460374 0.8393356 0.8326014 0.8258361 0.8190408 0.8122169 [17,] 0.8435490 0.8376843 0.8317797 0.8258361 0.8198543 0.8138354 0.8077803 [18,] 0.8344004 0.8293239 0.8242038 0.8190408 0.8138354 0.8085883 0.8033000 [19,] 0.8252618 0.8209579 0.8166095 0.8122169 0.8077803 0.8033000 0.7987764 [20,] 0.8161350 0.8125882 0.8089984 0.8053657 0.8016900 0.7979713 0.7942098 [21,] 0.8070220 0.8042166 0.8013721 0.7984885 0.7955654 0.7926029 0.7896007 [22,] 0.7979248 0.7958448 0.7937322 0.7915866 0.7894078 0.7871955 0.7849494 [23,] 0.7888453 0.7874748 0.7860803 0.7846615 0.7832181 0.7817498 0.7802563 [24,] 0.7797854 0.7791082 0.7784180 0.7777145 0.7769975 0.7762668 0.7755221 [25,] 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 [1,] 0.8553638 0.8377281 0.8204560 0.8035399 0.7869727 0.770747 [2,] 0.8523272 0.8353722 0.8187424 0.8024320 0.7864354 0.770747 [3,] 0.8492489 0.8329804 0.8170001 0.8013039 0.7858876 0.770747 [4,] 0.8461286 0.8305523 0.8152288 0.8001553 0.7853291 0.770747 [5,] 0.8429662 0.8280876 0.8134281 0.7989860 0.7847596 0.770747 [6,] 0.8397616 0.8255861 0.8115977 0.7977955 0.7841789 0.770747 [7,] 0.8365145 0.8230475 0.8097372 0.7965837 0.7835869 0.770747 [8,] 0.8332248 0.8204715 0.8078463 0.7953501 0.7829834 0.770747 [9,] 0.8298925 0.8178579 0.8059248 0.7940945 0.7823682 0.770747 [10,] 0.8265175 0.8152063 0.8039721 0.7928165 0.7817409 0.770747 [11,] 0.8230996 0.8125167 0.8019882 0.7915158 0.7811016 0.770747 [12,] 0.8196387 0.8097886 0.7999725 0.7901922 0.7804498 0.770747 [13,] 0.8161350 0.8070220 0.7979248 0.7888453 0.7797854 0.770747 [14,] 0.8125882 0.8042166 0.7958448 0.7874748 0.7791082 0.770747 [15,] 0.8089984 0.8013721 0.7937322 0.7860803 0.7784180 0.770747 [16,] 0.8053657 0.7984885 0.7915866 0.7846615 0.7777145 0.770747 [17,] 0.8016900 0.7955654 0.7894078 0.7832181 0.7769975 0.770747 [18,] 0.7979713 0.7926029 0.7871955 0.7817498 0.7762668 0.770747 [19,] 0.7942098 0.7896007 0.7849494 0.7802563 0.7755221 0.770747 [20,] 0.7904055 0.7865586 0.7826691 0.7787372 0.7747632 0.770747 [21,] 0.7865586 0.7834766 0.7803545 0.7771923 0.7739898 0.770747 [22,] 0.7826691 0.7803545 0.7780053 0.7756211 0.7732018 0.770747 [23,] 0.7787372 0.7771923 0.7756211 0.7740234 0.7723988 0.770747 [24,] 0.7747632 0.7739898 0.7732018 0.7723988 0.7715806 0.770747 [25,] 0.7707470 0.7707470 0.7707470 0.7707470 0.7707470 0.770747 > > > ##### > # (2) distribution function > > cbind(u, v, pfrank(u, v, 1/2)) u v [1,] 0.00000000 0.00000000 0.000000000 [2,] 0.04166667 0.04166667 0.002161918 [3,] 0.08333333 0.08333333 0.008483687 [4,] 0.12500000 0.12500000 0.018746096 [5,] 0.16666667 0.16666667 0.032761889 [6,] 0.20833333 0.20833333 0.050371846 [7,] 0.25000000 0.25000000 0.071441623 [8,] 0.29166667 0.29166667 0.095859213 [9,] 0.33333333 0.33333333 0.123532902 [10,] 0.37500000 0.37500000 0.154389663 [11,] 0.41666667 0.41666667 0.188373912 [12,] 0.45833333 0.45833333 0.225446581 [13,] 0.50000000 0.50000000 0.265584479 [14,] 0.54166667 0.54166667 0.308779915 [15,] 0.58333333 0.58333333 0.355040579 [16,] 0.62500000 0.62500000 0.404389663 [17,] 0.66666667 0.66666667 0.456866235 [18,] 0.70833333 0.70833333 0.512525880 [19,] 0.75000000 0.75000000 0.571441623 [20,] 0.79166667 0.79166667 0.633705179 [21,] 0.83333333 0.83333333 0.699428556 [22,] 0.87500000 0.87500000 0.768746096 [23,] 0.91666667 0.91666667 0.841817020 [24,] 0.95833333 0.95833333 0.918828585 [25,] 1.00000000 1.00000000 1.000000000 > cbind(u, v, outer(u, v, pfrank, alpha=1/2)) u v [1,] 0.00000000 0.00000000 0 0.000000000 0.000000000 0.000000000 0.000000000 [2,] 0.04166667 0.04166667 0 0.002161918 0.004281530 0.006359619 0.008396958 [3,] 0.08333333 0.08333333 0 0.004281530 0.008483687 0.012607780 0.016655103 [4,] 0.12500000 0.12500000 0 0.006359619 0.012607780 0.018746096 0.024776170 [5,] 0.16666667 0.16666667 0 0.008396958 0.016655103 0.024776170 0.032761889 [6,] 0.20833333 0.20833333 0 0.010394305 0.020626934 0.030699591 0.040613989 [7,] 0.25000000 0.25000000 0 0.012352407 0.024524535 0.036517939 0.048334193 [8,] 0.29166667 0.29166667 0 0.014271996 0.028349152 0.042232780 0.055924219 [9,] 0.33333333 0.33333333 0 0.016153796 0.032102017 0.047845666 0.063385780 [10,] 0.37500000 0.37500000 0 0.017998515 0.035784344 0.053358137 0.070720581 [11,] 0.41666667 0.41666667 0 0.019806850 0.039397334 0.058771720 0.077930323 [12,] 0.45833333 0.45833333 0 0.021579488 0.042942168 0.064087928 0.085016698 [13,] 0.50000000 0.50000000 0 0.023317101 0.046420017 0.069308261 0.091981390 [14,] 0.54166667 0.54166667 0 0.025020354 0.049832032 0.074434203 0.098826075 [15,] 0.58333333 0.58333333 0 0.026689897 0.053179351 0.079467226 0.105552419 [16,] 0.62500000 0.62500000 0 0.028326369 0.056463094 0.084408786 0.112162081 [17,] 0.66666667 0.66666667 0 0.029930399 0.059684368 0.089260326 0.118656709 [18,] 0.70833333 0.70833333 0 0.031502606 0.062844263 0.094023271 0.125037939 [19,] 0.75000000 0.75000000 0 0.033043596 0.065943855 0.098699036 0.131307399 [20,] 0.79166667 0.79166667 0 0.034553967 0.068984203 0.103289017 0.137466706 [21,] 0.83333333 0.83333333 0 0.036034303 0.071966353 0.107794596 0.143517462 [22,] 0.87500000 0.87500000 0 0.037485180 0.074891334 0.112217140 0.149461262 [23,] 0.91666667 0.91666667 0 0.038907165 0.077760160 0.116558000 0.155299686 [24,] 0.95833333 0.95833333 0 0.040300812 0.080573831 0.120818513 0.161034303 [25,] 1.00000000 1.00000000 0 0.041666667 0.083333333 0.125000000 0.166666667 [1,] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 [2,] 0.01039431 0.01235241 0.01427200 0.01615380 0.01799851 0.01980685 [3,] 0.02062693 0.02452454 0.02834915 0.03210202 0.03578434 0.03939733 [4,] 0.03069959 0.03651794 0.04223278 0.04784567 0.05335814 0.05877172 [5,] 0.04061399 0.04833419 0.05592422 0.06338578 0.07072058 0.07793032 [6,] 0.05037185 0.05997489 0.06942484 0.07872343 0.08787241 0.09687351 [7,] 0.05997489 0.07144162 0.08273603 0.09385974 0.10481440 0.11560167 [8,] 0.06942484 0.08273603 0.09585921 0.10879583 0.12154736 0.13411528 [9,] 0.07872343 0.09385974 0.10879583 0.12353290 0.13807215 0.15241482 [10,] 0.08787241 0.10481440 0.12154736 0.13807215 0.15438966 0.17050083 [11,] 0.09687351 0.11560167 0.13411528 0.15241482 0.17050083 0.18837391 [12,] 0.10572846 0.12622324 0.14650110 0.16656218 0.18640664 0.20603468 [13,] 0.11443901 0.13668077 0.15870636 0.18051554 0.20210808 0.22348382 [14,] 0.12300690 0.14697597 0.17073261 0.19427622 0.21760620 0.24072203 [15,] 0.13143386 0.15711053 0.18258142 0.20784558 0.23290209 0.25775008 [16,] 0.13972164 0.16708617 0.19425437 0.22122501 0.24799687 0.27456876 [17,] 0.14787197 0.17690460 0.20575307 0.23441592 0.26289168 0.29117891 [18,] 0.15588658 0.18656754 0.21707914 0.24741974 0.27758770 0.30758142 [19,] 0.16376721 0.19607672 0.22823420 0.26023793 0.29208617 0.32377720 [20,] 0.17151556 0.20543387 0.23921992 0.27287197 0.30638831 0.33976720 [21,] 0.17913337 0.21464073 0.25003794 0.28532338 0.32049541 0.35555242 [22,] 0.18662235 0.22369904 0.26068994 0.29759366 0.33440879 0.37113389 [23,] 0.19398420 0.23261052 0.27117760 0.30968437 0.34812976 0.38651268 [24,] 0.20122063 0.24137693 0.28150261 0.32159707 0.36165970 0.40168990 [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,] 0.02157949 0.02331710 0.02502035 0.02668990 0.02832637 0.02993040 [3,] 0.04294217 0.04642002 0.04983203 0.05317935 0.05646309 0.05968437 [4,] 0.06408793 0.06930826 0.07443420 0.07946723 0.08440879 0.08926033 [5,] 0.08501670 0.09198139 0.09882607 0.10555242 0.11216208 0.11865671 [6,] 0.10572846 0.11443901 0.12300690 0.13143386 0.13972164 0.14787197 [7,] 0.12622324 0.13668077 0.14697597 0.15711053 0.16708617 0.17690460 [8,] 0.14650110 0.15870636 0.17073261 0.18258142 0.19425437 0.20575307 [9,] 0.16656218 0.18051554 0.19427622 0.20784558 0.22122501 0.23441592 [10,] 0.18640664 0.20210808 0.21760620 0.23290209 0.24799687 0.26289168 [11,] 0.20603468 0.22348382 0.24072203 0.25775008 0.27456876 0.29117891 [12,] 0.22544658 0.24464264 0.26362321 0.28238869 0.30093953 0.31927622 [13,] 0.24464264 0.26558448 0.28630931 0.30681715 0.32710808 0.34718220 [14,] 0.26362321 0.28630931 0.30877991 0.33103468 0.35307330 0.37489551 [15,] 0.28238869 0.30681715 0.33103468 0.35504058 0.37883417 0.40241482 [16,] 0.30093953 0.32710808 0.35307330 0.37883417 0.40438966 0.42973882 [17,] 0.31927622 0.34718220 0.37489551 0.40241482 0.42973882 0.45686624 [18,] 0.33739928 0.36703970 0.39650110 0.42578195 0.45488069 0.48379583 [19,] 0.35530930 0.38668077 0.41788990 0.44893501 0.47981440 0.51052640 [20,] 0.37300690 0.40610568 0.43906179 0.47187351 0.50453908 0.53705677 [21,] 0.39049274 0.42531472 0.46001670 0.49459699 0.52905391 0.56338578 [22,] 0.40776754 0.44430826 0.48075459 0.51710505 0.55335814 0.58951233 [23,] 0.42483203 0.46308668 0.50127550 0.53939733 0.57745101 0.61543535 [24,] 0.44168702 0.48165043 0.52157949 0.56147352 0.60133185 0.64115380 [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,] 0.03150261 0.03304360 0.03455397 0.03603430 0.03748518 0.03890716 [3,] 0.06284426 0.06594386 0.06898420 0.07196635 0.07489133 0.07776016 [4,] 0.09402327 0.09869904 0.10328902 0.10779460 0.11221714 0.11655800 [5,] 0.12503794 0.13130740 0.13746671 0.14351746 0.14946126 0.15529969 [6,] 0.15588658 0.16376721 0.17151556 0.17913337 0.18662235 0.19398420 [7,] 0.18656754 0.19607672 0.20543387 0.21464073 0.22369904 0.23261052 [8,] 0.21707914 0.22823420 0.23921992 0.25003794 0.26068994 0.27117760 [9,] 0.24741974 0.26023793 0.27287197 0.28532338 0.29759366 0.30968437 [10,] 0.27758770 0.29208617 0.30638831 0.32049541 0.33440879 0.34812976 [11,] 0.30758142 0.32377720 0.33976720 0.35555242 0.37113389 0.38651268 [12,] 0.33739928 0.35530930 0.37300690 0.39049274 0.40776754 0.42483203 [13,] 0.36703970 0.38668077 0.40610568 0.42531472 0.44430826 0.46308668 [14,] 0.39650110 0.41788990 0.43906179 0.46001670 0.48075459 0.50127550 [15,] 0.42578195 0.44893501 0.47187351 0.49459699 0.51710505 0.53939733 [16,] 0.45488069 0.47981440 0.50453908 0.52905391 0.55335814 0.57745101 [17,] 0.48379583 0.51052640 0.53705677 0.56338578 0.58951233 0.61543535 [18,] 0.51252588 0.54106936 0.56942484 0.59759089 0.62556611 0.65334915 [19,] 0.54106936 0.57144162 0.60164155 0.63166753 0.66151794 0.69119120 [20,] 0.56942484 0.60164155 0.63370518 0.66561399 0.69736626 0.72896027 [21,] 0.59759089 0.63166753 0.66561399 0.69942856 0.73310950 0.76665510 [22,] 0.62556611 0.66151794 0.69736626 0.73310950 0.76874610 0.80427445 [23,] 0.65334915 0.69119120 0.72896027 0.76665510 0.80427445 0.84181702 [24,] 0.68093866 0.72068574 0.76039431 0.80006362 0.83969295 0.87928153 [25,] 0.70833333 0.75000000 0.79166667 0.83333333 0.87500000 0.91666667 [1,] 0.00000000 0.00000000 [2,] 0.04030081 0.04166667 [3,] 0.08057383 0.08333333 [4,] 0.12081851 0.12500000 [5,] 0.16103430 0.16666667 [6,] 0.20122063 0.20833333 [7,] 0.24137693 0.25000000 [8,] 0.28150261 0.29166667 [9,] 0.32159707 0.33333333 [10,] 0.36165970 0.37500000 [11,] 0.40168990 0.41666667 [12,] 0.44168702 0.45833333 [13,] 0.48165043 0.50000000 [14,] 0.52157949 0.54166667 [15,] 0.56147352 0.58333333 [16,] 0.60133185 0.62500000 [17,] 0.64115380 0.66666667 [18,] 0.68093866 0.70833333 [19,] 0.72068574 0.75000000 [20,] 0.76039431 0.79166667 [21,] 0.80006362 0.83333333 [22,] 0.83969295 0.87500000 [23,] 0.87928153 0.91666667 [24,] 0.91882858 0.95833333 [25,] 0.95833333 1.00000000 > > > > ##### > # (3) survival probabilities > > checkFrankfrechet <- function(x, omegatilde, beta) + { + v <- 1-exp(-1/(x*omegatilde)) + u <- 1-exp(-1/x) + u+v-1-1/beta*log(1- (1-exp(-beta*(1-u)))*(1-exp(-beta*(1-v)))/(1-exp(-beta))) + } > > > x <- 1:20 > p <- pfrechet(x, 1, 0) > pfrank(p, p, 1/2, lower.tail=FALSE) [1] 0.67737579 -0.04399087 -0.34226752 -0.50108736 -0.59892970 -0.66501956 [7] -0.71256676 -0.74837613 -0.77629782 -0.79866949 -0.81699080 -0.83226713 [13] -0.84519732 -0.85628204 -0.86588921 -0.87429521 -0.88171167 -0.88830332 [19] -0.89420030 -0.89950678 > checkFrankfrechet(x, 1, 1/2) [1] 0.413134675 0.169070453 0.090795103 0.056514207 0.038531805 0.027943890 [7] 0.021189036 0.016617672 0.013380816 0.011005342 0.009210634 0.007821704 [13] 0.006724832 0.005843518 0.005124765 0.004530915 0.004034617 0.003615618 [19] 0.003258658 0.002952069 > > > y <- 1:20 > p2 <- pfrechet(2*y, 1, 0) > pfrank(p, p2, 1/2, lower.tail=FALSE) [1] 0.2880794 -0.2879518 -0.5127563 -0.6306978 -0.7029971 -0.7517535 [7] -0.7868200 -0.8132369 -0.8338459 -0.8503688 -0.8639091 -0.8752063 [13] -0.8847744 -0.8929818 -0.9000990 -0.9063296 -0.9118293 -0.9167196 [19] -0.9210964 -0.9250364 > checkFrankfrechet(x, 2, 1/2) [1] 0.262489534 0.097379658 0.050256730 0.030599883 0.020571095 0.014772614 [7] 0.011120729 0.008673052 0.006952884 0.005698048 0.004754658 0.004027559 [13] 0.003455348 0.002996961 0.002624093 0.002316721 0.002060357 0.001844308 [19] 0.001660543 0.001502936 > > > ##### > # (4) simulation > > n <- 1e5 > > uv <- rfrank(n, 1/2) > S <- function(x, y) sum(uv[,1] > x & uv[,2] > y) / NROW(uv) > > S(1/2, 1/4) [1] 0.38772 > pfrank(1/2, 1/4, 1/2, lower.tail=FALSE) [1] 0.6366808 > > S(1/2, 1/2) [1] 0.26758 > pfrank(1/2, 1/2, 1/2, lower.tail=FALSE) [1] 0.2655845 > > S(1/2, 3/4) [1] 0.13821 > pfrank(1/2, 3/4, 1/2, lower.tail=FALSE) [1] -0.1133192 > > > proc.time() user system elapsed 0.26 0.14 0.39