ACTU-457 (Mast 724) Risk Theory

library(scatterplot3d)


n <- 1000
l1 <- 1
l2 <- 2
X1 <- rexp(n,l1)
X2 <- rexp(n,l2)
U1 <- pexp(X1,l1)
U2 <- pexp(X2,l2)
Indep <- matrix(c(X1,X2,U1*U2),ncol=3)
if(require(scatterplot3d))
par(mfrow=c(2,2))
fig1 <- scatterplot3d(Indep)

#comonotonic case (perfect positive dependence)
l1 <- 1
l2 <- 2
X1 <- rexp(n,l1)
X2 <- rexp(n,l2)
U1 <- pexp(X1,l1)
U2 <- pexp(X2,l2)
copula <- min(U1,U2)
Comono <- matrix(c(X1,X2,copula),ncol=3)
if(require(scatterplot3d))
scatterplot3d(Comono)

#Clayton example
l1 <- 1
l2 <- 2
tt <- -0.5
X1 <- rexp(n,l1)
X2 <- rexp(n,l2)
U1 <- pexp(X1,l1)
U2 <- pexp(X2,l2)
copula <- (U1^(-tt)+U2^(-tt)-1)^(-1/tt)
Clayton <- matrix(c(X1,X2,copula),ncol=3)
if(require(scatterplot3d))
scatterplot3d(Clayton)



library(copula)

norm.cop <- normalCopula(0.5)
norm.cop

Normal copula, dim. d = 2 
Dimension:  2 
Parameters:
  rho.1   = 0.5

## one d-vector =^= 1-row matrix, works too :
dCopula(c(0.5, 0.5), norm.cop)

[1] 1.154701

pCopula(c(0.5, 0.5), norm.cop)

[1] 0.3333333

u <- rCopula(n, norm.cop)
plot(u)

dCopula(u, norm.cop)

   [1]  2.77969757  1.20871444  0.20932848  1.29037083  1.56510294  1.07933251
   [7]  2.01281368  1.22193954  1.16659427  0.90265543  1.19016671  0.99659721
  [13]  0.97622012  2.73641208  1.22639224  0.99412900  0.67140321  1.42349100
  [19]  0.93746346  1.23571576  1.20295335  2.59149222  1.43061658  5.44325043
  [25]  1.22278117  1.22305469  1.07595197  1.14169071  1.28970487  1.24389353
  [31]  1.90095646  1.20278164  1.80034650  1.15652130  1.75258093  1.91676225
  [37]  1.43705223  0.95265574  0.85474863  0.89484636  1.36191922  1.10171733
  [43]  1.10929924  1.16572189  2.25552563  1.02299745  1.25003119  3.68299640
  [49]  1.18348099  1.38168953  1.22982680  2.28755868  1.19400369  1.13784403
  [55]  1.13863308  1.12088730  1.54289964  1.48849345  1.46917301  1.24315973
  [61]  1.33601080  1.32696709  0.74031502  0.92561433  3.59133622  1.16606359
  [67]  1.24412610  1.01552959  0.70953129  1.64469101  1.35796587  1.05858619
  [73]  0.88729423  1.39973000  0.96275942  1.28355039  1.18967068  1.20073049
  [79]  0.80328842  1.33250820  1.66768576  1.19790120  2.19757861  1.19350095
  [85]  1.74531801  1.15136540  1.16138241  1.47597167  1.12340433  0.86171156
  [91]  1.03800765  1.16024476  2.05768899  1.80377624  0.89803545  2.22014654
  [97]  0.14478608  0.94204502  2.00432739  0.83917187  0.66720632  2.13455761
 [103]  0.59172486  0.70688658  1.15872220  1.09582817  0.82239755  0.91878547
 [109]  2.01986054  1.17753966  0.22251638  0.88439148  2.28364799  0.62844439
 [115]  1.32650841  1.65407032  1.20628986  1.88681847  1.94995557  0.91149899
 [121]  0.58479678  1.63333289  1.03407626  0.77242331  1.07484549  1.16553848
 [127]  1.55247482  1.21497260  0.36500688  0.83473712  1.19543439  0.56096764
 [133]  1.23764004  3.20316879  0.30991853  1.63009176  0.95076855  0.70304148
 [139]  0.86194554  1.07401694  3.06574870  0.67397608  0.67042598  0.37925764
 [145]  1.70427258  1.15196786  1.05405530  1.52926528  0.96236022  0.43650111
 [151]  0.95455427  1.20777523  1.50403815  0.84903569  0.96939661  1.17063441
 [157]  1.03370930  0.76128946  1.12019147  4.39161497  1.22270285  1.16204602
 [163]  1.04495228  1.18075555  1.56804861  1.43408348  1.01321865  0.92852664
 [169]  1.36765911  2.41611968  0.67612013  1.13421259  1.09254274  1.20728878
 [175]  0.66705303  1.13798012  0.54994257  1.14481655  1.62648352  1.21439823
 [181]  1.15706973  0.99993102  1.18831791  1.24801826  0.15675655  0.48057101
 [187]  0.69959809  0.64900173  1.23268666  1.21738730  1.10920573  0.75926444
 [193]  2.29268496  1.70376671  1.69009471  1.32917028  1.30902065  1.64861437
 [199]  1.01383459  1.50541292  0.67321648  1.28395912  2.26221105  1.24528675
 [205]  1.22327188  1.23798099  1.26867053  2.86118257  0.34261695  1.36525136
 [211]  1.33631016  1.13292711  1.62019129  2.33302236  0.30204836  2.99968498
 [217]  1.33245742  0.38879664  0.79224865  1.19664401  0.85976997  1.02329175
 [223]  0.95455793  0.79907780  0.48094907  0.72150051  1.02051534  0.63821355
 [229]  0.98122402  1.36643479  1.27142103  0.93857066  2.66922502  0.85909059
 [235]  0.25537029  1.05257924  1.84425933  1.42773218  1.21371947  0.99505296
 [241]  1.38049101  1.11551126  1.66238035  2.45187080  0.58843709  1.12288446
 [247]  0.97801547  0.82386368  0.33245463  1.72085788  1.26763544  1.06284572
 [253]  1.13054819  0.85329526  2.04474154  0.65681587  1.04134876  1.15591392
 [259]  1.92197568  1.88392636  0.90951072  1.09058957  2.13407611  2.04183698
 [265]  1.35830248  1.12899579  1.10165048  0.90883498  1.15387985  1.36136026
 [271]  1.12181099  3.35736133  1.12776395  1.16892340  1.11845027  0.94533042
 [277]  1.31313782  1.15594797  0.91355060  1.54232593  1.18070868  1.23022214
 [283]  0.98969090  1.96298101  1.50595508  0.65362650  0.72435856  1.10207826
 [289]  1.29061946  1.06621181  1.17236302  1.07544541  1.07009494  1.90511868
 [295]  2.60394277  1.15921782  1.28303858  1.42367107  0.93619780  1.26638602
 [301]  1.34571543  1.24341813  0.94896304  1.03832101  1.04409714  1.21670710
 [307]  1.16127384  0.93816611  1.30941072  1.20347155  1.13232586  1.08703046
 [313]  1.00583631  1.25863680  0.95293937  1.30120852  1.31323829  0.86285549
 [319]  0.53796240  1.04044351  5.39053074  1.33516742  1.29760499  1.05931239
 [325]  1.33020724  1.21468366  1.13601199  0.41413547  0.94797690  1.14423227
 [331]  1.97911137  0.26534787  1.27979754  3.43228262  1.02075163  2.53237443
 [337]  0.50718840  1.15458328  0.53656119  1.66271709  1.10762582  1.17745269
 [343]  1.55708323  0.55375877  0.94902672  1.19323347  2.17716630  0.51924966
 [349]  2.31367856  1.02296674  1.24042654  1.17435522  1.12446732  0.83098403
 [355]  1.16989991  1.19729446  0.81583526  0.90966908  1.55693962  0.99139946
 [361]  5.75654385  1.34967108  1.20195288  1.11912215  0.77895366  1.41880252
 [367]  1.70608774  1.62697724  0.98749704  1.33075722  0.83323807  1.14615751
 [373]  1.46133133  1.21343305  1.17363303  0.06897765  0.69427011  1.14589311
 [379]  1.14998027  1.05096266  1.03834954  1.07417502  1.48194396  1.08109384
 [385]  2.09192854  3.25149220  1.16556812  2.69230930  1.32710400  1.44458675
 [391]  1.02673071  0.63199770  3.10134508  0.92782653  1.16022081  0.79371100
 [397]  1.53962863  0.51847944  1.21618277  1.76135546  1.15342672  1.57098669
 [403]  1.92223290  1.15295968  1.55175334  1.21398645  1.22443273  4.61478870
 [409]  1.57449510  1.80238758  1.15765457  1.13341943  0.88714394  1.31317044
 [415]  0.67147273  1.25376115  1.57346483  1.21287561  1.02476553  1.15378310
 [421]  0.95034698  1.07047249  0.95539797  1.29324410  0.54420970  2.65393942
 [427]  1.11016409  1.14372386  1.23808657  1.42246401  1.17514117  1.20384316
 [433]  1.14454022  1.40920150  1.05089355  0.38769241  1.19602189  1.18097047
 [439]  1.25552358  1.26469103  1.14969339  1.11469454  2.23163139  1.16332389
 [445]  1.22527054  1.42812384  1.54654058  1.75239728  1.13727532  0.52847277
 [451]  0.90096727  1.69298730  0.60155092  1.14881816  1.10238906  1.22987407
 [457]  1.22388708  0.52854817  0.96507655  0.79663544  1.51716610  6.11570195
 [463]  1.15313970  1.35837125  1.99059773  1.30607950  2.47187358  1.05361074
 [469]  0.98852150  0.44501172  1.14186615  7.42814503  2.16932630  0.76265133
 [475]  1.39838234  1.70015134  0.70060665  1.43482038  1.07756174  1.94124010
 [481]  1.12743406  0.89943128  1.15262024  1.25186217  1.14948907  1.38242908
 [487]  1.17931302  1.41276702  0.89336781  1.17642271  1.45255893  2.01585006
 [493]  2.50753901  1.33099707  1.25454972  0.84793948  1.23917277  0.96793071
 [499]  1.74415716  0.68823929  0.96369832  0.88948767  0.74567202  0.86995926
 [505]  1.32553493  1.41069478  0.25856673  0.70980408  1.54808792  0.34757684
 [511]  1.13835092  1.34250618  0.30166874  0.69504518  0.96829857  0.83459885
 [517]  0.84068831  0.73022544  0.89274322  2.81791317  1.13088721  1.21323042
 [523]  1.16584784  1.18581012  1.08787984  0.58633840  3.48777103  1.00114655
 [529]  1.79770008  0.84404564  1.07102541  2.04347071  2.73325926  1.08708227
 [535]  0.43234029  1.49399869  1.10216148  1.69330692  1.65305318  0.46718153
 [541]  1.36653964  2.53017610  1.35336012  1.25787684  1.44548260  1.01748054
 [547]  8.25025191  1.25033789  0.88587275  0.73115582  1.12790270  4.30516898
 [553]  1.17481860  1.13730792  1.15337417  0.72399646  0.84963903  1.40240698
 [559]  1.69333562  0.24274432  0.81549584  0.95706150  0.98548146  0.56396362
 [565]  1.06360544  0.95356000  1.41854526  1.18658830  1.37291817  1.11710809
 [571]  0.46782866  0.99323667  1.69627707  1.20617416  1.18155398  0.94179940
 [577]  1.02811930  1.46912989  0.65436537  2.05852823  1.17751319  0.75444040
 [583]  1.08652635  2.96154346  1.21388861  1.51037325  0.36768739  1.17142929
 [589]  0.91555094  1.23071195  0.70345551  0.62702504  0.90586597  0.53933924
 [595]  1.10380737  2.02795726  1.13795782  1.93574386  1.12347853  1.03594172
 [601]  1.10445666  3.75036715  3.34883335  0.94468324  1.02143827  1.23254982
 [607]  0.52126343  0.88750140  1.09086375  0.80061498  1.32445713  0.95407430
 [613]  0.84760660  1.14056102  1.43189346  1.14161991  0.91311068  1.00021690
 [619]  1.09775855  1.86297196  1.48963116  1.15780814  2.51809100  1.16907979
 [625]  0.75626501  0.71522034  3.05222939  0.82489071  1.22065042  0.85482614
 [631]  1.45330325  1.53986839  1.00442773  0.77600966  0.63287494  1.12565486
 [637]  1.25423772  1.38425076  1.54332197  0.62306299  0.41063991  0.91015715
 [643]  1.10221901  0.94546715  0.25901170  1.12573417  1.05968656  1.14809920
 [649]  1.74605755  0.47012424  2.57215187  1.26370347  0.80459553  0.62917479
 [655]  2.24190669  0.74867732  1.37837425  1.14869480  1.33271332  1.57255338
 [661]  1.87959744  0.93638668  1.17277734  1.47057980  1.94480934  1.00784428
 [667]  1.23049664  1.11669961  1.89018287  0.95315424  6.37323862  1.09609611
 [673]  1.16051604  0.68607620  1.15736421  2.41957917  1.13878111  1.67627089
 [679]  0.84351531  1.75301971  1.32125904  0.64651241  0.85848997  0.18924307
 [685]  0.78298833  0.61764163  0.96655072  0.46332040  1.63361309  1.12487865
 [691]  1.28623539  1.04173026  1.54752410  1.42878163  1.19074307  1.13998971
 [697]  1.20542625  0.92651049  1.56273126  0.45139407  0.35346607  1.29340922
 [703]  0.77029043  1.02333548  0.96338920  0.91655353  1.26569577  1.21566898
 [709]  1.70879089  0.73225550  0.88547259  1.35254126  0.74484369  1.09564286
 [715]  0.72813061  1.40949662  1.05144570  1.33332385  2.31868970  1.45552753
 [721]  0.28206203  1.27179695  1.36881910  1.36908619  1.07899178  1.75388958
 [727]  0.74669250  1.44818283  1.10164739  1.27613275  1.16244104  1.59372438
 [733]  0.89526635  1.11964108  0.94894856  1.12189219  1.32354587  5.30566347
 [739]  1.26711389  0.86154100  1.22982532  2.03326465  0.38679909  1.58926921
 [745]  0.50978853  0.98468401  1.16042887  1.35781709  0.35910367  1.17912163
 [751]  1.78258125  0.60164446  1.10991036  1.98202536  1.04045524  1.11448157
 [757]  1.14940157  1.51702639  0.54623888  1.01768853  1.31402546  1.07821097
 [763] 40.02691121  0.78595032  0.68504066  1.02896319  1.15330430  1.17580231
 [769]  1.12274518  1.22571587  1.69551731  1.53374105  0.94559218  1.22050095
 [775]  0.22452591  1.08540077  0.98559061  1.43481005  1.07490849  1.43708849
 [781]  1.49576885  2.22703015  0.98742654  0.55473988  1.09276241  1.11903718
 [787]  1.14628435  0.34091809  1.18152095  1.37293740  1.45463947  1.19612266
 [793]  1.12209784  0.35910734  1.46933191  0.95600456  1.24444336  0.89836106
 [799]  1.44766540  0.87240328  1.08640789  1.22342806  1.76983486  0.84120501
 [805]  0.88378475  1.50248176  1.21978664  1.01132628  1.07137802  1.02399236
 [811]  1.10633382  1.14785883  2.24722590  1.13441216  1.13750886  0.71934651
 [817]  0.96460060  1.15383359  1.39185778  1.14748764  1.16926565  0.58309954
 [823]  2.34263487  0.59189470  0.81425248  0.59626116  2.87651602  1.39195972
 [829]  1.14137356  1.20793511  1.21399473  1.02031037  1.43288425  0.96296420
 [835]  1.41834136  1.12543593  0.99283432  1.15172711  1.11098981  1.58146949
 [841]  0.75764233  0.85230806  1.28811162  0.84879595  1.16460250  1.10624694
 [847]  2.79057090  0.67694484  1.08975014  2.78574999  0.58239530  0.55692645
 [853]  1.06661852  1.15022260  2.79103018  0.96672182  0.53037565  1.19260523
 [859]  1.15643833  0.94956020  0.24982044  0.73241246  0.94414474  0.65456630
 [865]  5.48789134  0.57018887  0.56577228  2.09124823  0.75130671  1.14029797
 [871]  3.78254886  1.03005890  5.11991341  0.85558797  1.61587502  0.89429592
 [877]  1.06520106  0.55602512  1.11889703  0.96072558  1.61458143  0.94784168
 [883]  1.19075540  0.92810840  0.90113948  1.33170260  1.13292956  1.11069386
 [889]  0.63786347  0.95461738  1.61887774  0.74989472  1.17160062  1.10846868
 [895]  1.30908603  2.29212823  1.13135557  1.27758979  0.32436977  1.14529097
 [901]  1.02868882  1.41071280  1.02231822  1.10147726  1.66662788  1.08503243
 [907]  0.95587731  0.36184906  1.11541868  1.30172373  0.76699755  1.33170476
 [913]  1.18305948  1.01001822  0.53402130  1.17005252  1.62484911  0.94509220
 [919]  1.11056033  0.53075402  2.03214229  1.13107488  0.58328052  1.35696463
 [925]  1.28157062  1.30303696  0.98868129  1.47564303  1.22571535  1.85392303
 [931]  1.18046378  0.96586157  1.13705851  1.16626104  1.49545089  1.22889375
 [937]  1.56844771  1.42556013  0.44913510  0.53126749  1.23762484  1.23063798
 [943]  1.14293027  1.18896690  1.07161412  1.02383070  1.13184112  1.68560486
 [949]  1.57803515  3.31562651  1.99001943  3.78215614  1.46253902  6.66363176
 [955]  1.22727976  1.30984842  1.15302096  1.11442681  0.44164364  1.43550294
 [961]  1.59960708  0.73505490  0.31564236  1.15017797  0.92992038  1.21951342
 [967]  1.02965959  1.32363031  0.97020922  1.36824838  0.68304922  0.88261554
 [973]  1.33590775  1.04487560  1.24495728  1.38559289  1.38887148  0.54411820
 [979]  0.64330269  1.15259685  1.07462465  0.75123138  0.95011504  0.90340750
 [985]  1.44389796  1.46118911  1.68321438  1.00767666  1.01629862  0.82479083
 [991]  1.46469671  0.43750979  1.08662326  0.99372715  1.28795762  1.31395328
 [997]  1.17416305  0.98995827  1.11176173  1.09006134

pCopula(u, norm.cop)

   [1] 9.055457e-01 1.493363e-01 2.355613e-01 7.228710e-01 7.352776e-01 2.332423e-01
   [7] 8.377604e-01 5.749740e-01 2.339516e-01 4.837571e-01 5.616725e-01 2.921658e-01
  [13] 1.377766e-01 9.048727e-01 5.137597e-01 4.589918e-01 1.995227e-01 7.103918e-01
  [19] 5.516266e-01 3.522164e-02 1.777019e-01 1.500785e-03 6.734840e-01 9.691848e-01
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 [967] 2.960062e-01 1.082114e-01 5.813804e-01 7.440451e-02 1.294801e-01 5.046213e-01
 [973] 6.228118e-01 1.332760e-01 6.514171e-01 9.808488e-02 8.809372e-02 6.352028e-02
 [979] 1.192136e-01 1.904173e-01 5.477199e-01 1.360254e-01 4.028805e-01 4.994055e-01
 [985] 7.276962e-01 3.305985e-02 7.703350e-01 2.151704e-01 3.482507e-01 8.506260e-02
 [991] 3.179623e-02 2.114379e-01 4.578711e-01 6.491603e-01 9.286277e-02 1.003440e-01
 [997] 4.616247e-01 3.388365e-01 5.428122e-01 5.968400e-01

persp (norm.cop, dCopula)
contour(norm.cop, pCopula)

## a 3-dimensional normal copula
u <- rCopula(1000, normalCopula(0.5, dim = 3))
if(require(scatterplot3d))
scatterplot3d(u)

 ### data location
mydata <-read.csv(file="/Users/mmailhot/Dropbox/Mac (2)/Desktop/DataCopula.csv",header = TRUE)
# Estimate x  gamma distribution parameters and visually compare simulated vs observed data
x_mean <- mean(mydata$x)
x_var <- var(mydata$x)
x_rate <- x_mean / x_var
x_shape <- ( (x_mean)^2 ) / x_var

hist(mydata$x, breaks = 20, col = "green", density = 20)
hist(rgamma( nrow(mydata), rate = x_rate, shape = x_shape), breaks = 20,col = "blue", add = T, density = 20, angle = -45)

# Estimate y gamma distribution parameters and visually compare simulated vs observed data
y_mean <- mean(mydata$y)
y_var <- var(mydata$y)
y_rate <- y_mean / y_var
y_shape <- ( (y_mean)^2 ) / y_var
hist(mydata$y, breaks = 20, col = "green", density = 20)
hist(rgamma(nrow(mydata), rate = y_rate, shape = y_shape), breaks = 20, col = "blue", add = T, density = 20, angle = -45)

# Measure association using Kendall's Tau
cor(mydata, method = "kendall")

           X          x          y
X 1.00000000 0.01748549 0.02908509
x 0.01748549 1.00000000 0.44954955
y 0.02908509 0.44954955 1.00000000

library(copula)
mydata <- matrix(c(mydata$x,mydata$y),ncol = 2)
# Estimate copula parameters
cop_model <- claytonCopula(dim = 2)
m <- pobs(as.matrix(mydata))
fit <- fitCopula(cop_model, m, method = 'ml')
coef(fit)

   alpha 
1.646685 

##    param 
## 1.482168 

# Check Kendall's tau value for the Clayton copula with theta = 1.48
tau(claytonCopula(param = 1.646685))

[1] 0.4515567

## [1] 0.4515567


# Build the bivariate distribution
my_dist <- mvdc(claytonCopula(param = 1.48, dim = 2), margins = c("gamma","gamma"), paramMargins = list(list(shape = x_shape, rate = x_rate), list(shape = y_shape, rate = y_rate)))

# Generate random sample observations from the multivariate distribution
v <- rMvdc(5000, my_dist)
# Compute the density
pdf_mvd <- dMvdc(v, my_dist)
# Compute the CDF
cdf_mvd <- pMvdc(v, my_dist)

install.packages("scatterplot3d") # Install

trying URL 'https://cran.rstudio.com/bin/macosx/big-sur-arm64/contrib/4.4/scatterplot3d_0.3-44.tgz'
Content type 'application/x-gzip' length 348854 bytes (340 KB)
==================================================
downloaded 340 KB

The downloaded binary packages are in
    /var/folders/xq/4d6jckb55r1fl1nqmql87cznpwjmt5/T//RtmprohrGQ/downloaded_packages

library("scatterplot3d") # load
# 3D plain scatterplot of the generated bivariate distribution
par(mfrow = c(1, 2))

scatterplot3d(v[,1],v[,2], pdf_mvd, color="red", main="Density", xlab = "u1", ylab="u2", zlab="pMvdc",pch=".")
scatterplot3d(v[,1],v[,2], cdf_mvd, color="red", main="CDF", xlab = "u1", ylab="u2", zlab="pMvdc",pch=".")

persp(my_dist, dMvdc, xlim = c(-4, 4), ylim=c(0, 2), main = "Density")
contour(my_dist, dMvdc, xlim = c(-4, 4), ylim=c(0, 2), main = "Contour plot")

persp(my_dist, pMvdc, xlim = c(-4, 4), ylim=c(0, 2), main = "CDF")
contour(my_dist, pMvdc, xlim = c(-4, 4), ylim=c(0, 2), main = "Contour plot")