library(MASS)#Question 1, Monte Carlo Bivariate T-Copula
set.seed(1)# Set seed for number sequence
dof<-10 #Degrees of freedom for the t-copula
Ndim<-3#the number of risks, rsik1,rsik2 and rsik3
rho<-0.4#T-copula correlation parameter
CoRMatrix<-matrix(c(1,rho,rho,rho,rho,rho,rho,rho,1),Ndim,Ndim,Ndim)# the correlation matrix formed from the 3 risks and 10 degrees of freedom of the t-copula
sigma<-c(log(c(1.4,1.7,2.0))) #matrix for the standard devistions. We use logs since the distribution is a log Normal
Nsim<-10000
Z<-mvrnorm(Nsim,rep(0,Ndim), CoRMatrix)# Estimates Z
EZ<-mean(Z)#Ouputs Question 1 part 1, the value of E[ Z]
EZ
var(Z)#Outputs Question 1a, part ii, VaR0:99[Z]
Z #Outputs Question 1a part iii, ES0:99[Z].
n=length(Z)
m=0 #both n and m will be used in calculating the 0.99 confidence interval
con.level<-0.99 # this is the level of confidence
Zed<-sqrt(0.995) # this the z value for calculating confidence interval
T<-Zed/sqrt(n)
se<-sd(T) # output the standard error
CI<-0.99*se #confidence interval
LowerLimit<-m-CI
UpperLimit<-m+CI
#########
####Question 1 B
cat(“MLE estimate of E[Z1]=”,mean(Z[,1]),”n”)
cat(“MLE estimate of E[Z2]=”,mean(Z[,2]),”n”)
cat(“MLE estimate of E[Z3]=”,mean(Z[,3]),”n”)
cat(“MLE estimate of standard deviation of Z1=”,sd(Z[,1]),”n”)
cat(“MLE estimate of standard deviation of Z2=”,sd(Z[,2]),”n”)
cat(“MLE estimate of standard deviation of Z3=”,sd(Z[,3]),”n”)
#########
####Question 1 C
##Kendall’s Correlation for measuring the strength of association between the variables. Correlation is between 2 variables hence we calculate for each pair
cat(“Estimate of kendall’s correlation between Z1 and Z2=”,cor(Z[,1],Z[,2],method=”kendall”), “n”)
cat(“Estimate of kendall’s correlation between Z1 and Z3=”,cor(Z[,1],Z[,3],method=”kendall”), “n”)
cat(“Estimate of kendall’s correlation between Z2 and Z3=”,cor(Z[,2],Z[,3]
,method=”kendall”), “n”)
####Spearsman’s Correlation also measures the strength of association between two variables hence we calculate for each pair
cat(“Estimate of Spearma’s correlation between Z1 and Z2=”,cor(Z[,1],Z[,2],method=”spearman”), “n”)
cat(“Estimate of Spearman’s correlation between Z1 and Z3=”,cor(Z[,1],Z[,3],method=”spearman”), “n”)
cat(“Estimate of Spearman’s correlation=”,cor(Z[,2],Z[,3],method=”spearman”), “n”)
####Question 1 d
##
U.tcopula<-pt(Z,dof)#gennerates a sample (U1, U2) from the t- copula
U.Gaussiancopula<-pnorm(Z,0,1) #generates sample (U1,U2) from Gaussian Copula
> library(MASS)#Question 1, Monte Carlo Bivariate T-Copula
> set.seed(1)# Set seed for number sequence
> dof<-10 #Degrees of freedom for the t-copula
> Ndim<-3#the number of risks, rsik1,rsik2 and rsik3
> rho<-0.4#T-copula correlation parameter
> CoRMatrix<-matrix(c(1,rho,rho,rho,rho,rho,rho,rho,1),Ndim,Ndim,Ndim)# the correlation matrix formed from the 3 risks and 10 degrees of freedom of the t-copula
> sigma<-c(log(c(1.4,1.7,2.0))) #matrix for the standard devistions. We use logs since the distribution is a log Normal
> Nsim<-10000
> Z<-mvrnorm(Nsim,rep(0,Ndim), CoRMatrix)# Estimates Z
> EZ<-mean(Z)#Ouputs Question 1 part 1, the value of E[ Z]
> EZ
[1] 0.004449141
> var(Z)#Outputs Question 1a, part ii, VaR0:99[Z]
[,1] [,2] [,3]
[1,] 1.0162244 0.4117166 0.4229360
[2,] 0.4117166 0.4085383 0.4085549
[3,] 0.4229360 0.4085549 1.0076161
> Z #Outputs Question 1a part iii, ES0:99[Z].
[,1] [,2] [,3]
[1,] 0.9861174186 2.494692e-01 0.1050162967
[2,] 0.4534764159 -1.820795e-01 -0.7038894508
[3,] 1.1890447540 6.647049e-01 0.0548255003
[4,] -0.8877186086 -1.716365e-01 -2.1864349065
[5,] 0.1157733377 -5.348002e-01 -0.4324306808
[6,] 0.9365952471 5.321820e-01 0.3614989325
[7,] -0.3945724223 2.317379e-01 -0.7258711271
[8,] -0.7907853384 -6.413838e-01 -0.2737701533
[9,] -0.1167143098 -1.009505e+00 -0.3888052931
[10,] -0.3556636452 -9.216153e-02 1.0238501385
[11,] -1.3486615794 -1.104422e+00 -0.9642469235
[12,] -0.7347963565 -3.986172e-01 0.2109217273
[13,] 0.7460218493 4.272179e-01 0.2214007953
[14,] 2.4129771118 1.405025e+00 1.1110633579
[15,] -0.8649885710 -3.462986e-01 -1.1591882928
[16,] 0.0512678656 1.176738e-01 -0.0366098979
[17,] 0.1155564246 -1.000934e-01 -0.0194415051
[18,] -0.5434974102 -1.078669e+00 -0.6524282596
[19,] -0.1251018787 -7.439065e-01 -1.0395290469
[20,] -1.6224439481 3.858225e-02 0.4126515771
[21,] -0.4504477838 -6.071954e-01 -0.9964143642
[22,] 0.1343519455 -6.875284e-01 -1.2569231706
[23,] 0.5175397849 -2.470938e-01 -0.5088306279
[24,] 1.6463028972 1.459214e-01 2.2306563096
[25,] 0.2845703586 -5.490302e-02 -1.4865048874
[26,] 0.2612151331 -9.898811e-02 -0.0861030137
[27,] 0.3678974156 5.481286e-02 -0.0919306520
[28,] 1.1948009994 1.166460e+00 0.9966833700
[29,] -0.0703004581 3.395729e-01 0.8080397170
[30,] 0.1273972715 -4.125270e-01 -0.6984784141
[31,] -1.2507842970 -6.346667e-01 -1.0560418715
[32,] -0.1660870190 4.848480e-01 0.0621411372
[33,] -0.6974023556 -4.057219e-01 0.1823903942
[34,] 0.8750889413 1.639643e-01 -0.8722352811
[35,] 0.2589037308 7.498773e-01 2.0111598338
[36,] 0.0538421460 2.722704e-01 0.6007657221
[37,] 0.1503261532 1.309845e-01 0.5530272183
[38,] 0.3717352630 -2.751404e-01 -0.0779779973
[39,] -1.4354164306 -5.337674e-01 -0.4195549441
[40,] -1.3397826211 3.286976e-01 -0.3927527055
[41,] -0.0516126742 -5.640009e-01 0.7394655201
[42,] 0.0435693178 1.903135e-01 0.3407258431
[43,] -0.7018876832 -5.195360e-01 -0.3577994496
[44,] -0.6498897622 -1.002010e+00 0.1777450374
[45,] 0.6926466879 3.081159e-01 0.4854326836
[46,] 0.3073162204 6.024017e-01 0.7205648776
[47,] 0.0548373538 -6.118715e-01 -0.3923594317
[48,] -0.4044836076 -6.796340e-01 -0.6959729086
[49,] 0.1101910883 -1.422543e-01 0.2046898333
[50,] -0.7536996562 -5.253960e-02 -0.9711594205
[51,] -0.5910085212 -3.562418e-01 0.0236322922
[52,] 0.1348510300 7.244603e-01 0.6247039200
[53,] -0.0385736308 -4.871161e-01 -0.3316532662
[54,] 0.9501446205 1.093916e+00 0.6063019015
[55,] -0.8059296064 -9.079927e-01 -1.4750247706
[56,] -1.7455370914 -9.231347e-01 -1.6181164811
[57,] 0.7849993186 6.854961e-02 -0.0958645450
[58,] 0.9027511188 7.811673e-01 0.6829767091
[59,] -0.6398686207 -3.585422e-01 -0.2685160389
[60,] -0.2594971036 6.557416e-01 0.1110058395
[61,] -1.6438009007 -1.142315e+00 -2.4207228027
[62,] -0.0579835200 -5.939347e-01 0.5149001853
[63,] -0.3274897390 -5.240733e-01 -0.7148887373
[64,] 0.3710579969 8.525273e-02 -0.4812845812
[65,] 0.3260007144 -6.607009e-02 1.1994026408
[66,] 0.3389612339 -9.852651e-02 -0.6529112138
[67,] 2.0351046949 9.170053e-01 0.9823430783
[68,] -0.9901267800 -1.039332e+00 -1.2720240719
[69,] -0.2667324020 -3.489143e-01 0.1833163012
[70,] -1.6282318005 -1.014161e+00 -2.0609766538
[71,] -1.0181445230 -1.536798e-01 0.1671755564
[72,] 0.7348652941 -1.730445e-01 0.7922224174
[73,] -1.3504497784 1.685272e-01 0.0242465936
[74,] 0.7676404536 4.778956e-01 0.7918191815
[75,] 0.3158141502 1.022237e+00 1.5343203205
[76,] -0.1963131609 3.190889e-01 -0.5887069911
[77,] -0.6080821051 -2.636346e-02 1.5095271822
[78,] 0.1677794343 -3.363765e-01 0.0444407971
[79,] -0.6037537798 2.317131e-01 0.3076896657
[80,] 0.0540002283 -1.065663e-02 1.1292491779
[81,] 0.0702482369 -2.292012e-01 1.2107013349
[82,] 0.1179220183 1.491830e-01 0.0567443300
[83,] -1.1347301895 -4.298866e-01 -0.9422417541
[84,] 1.5349640029 1.519146e+00 0.5370943065
[85,] 0.4115783509 -5.169401e-01 -1.2673377544
[86,] -1.0087396264 3.287866e-01 0.1347113522
[87,] -1.5915513338 -5.122122e-01 -0.2034715044
[88,] 0.6574667263 1.710231e-01 -0.1594531568
[89,] -0.1637689015 -2.757378e-01 -0.3988743621
[90,] 0.3550569878 5.258569e-01 -1.2232951007
[91,] -0.1616928936 4.245904e-01 0.9736946608
[92,] -1.0617025994 -7.708415e-01 -0.8573557463
[93,] -0.9422019773 -6.485094e-01 -0.9601157844
[94,] 0.0135205780 -3.061104e-01 -1.2157440386
[95,] -1.9423445148 -9.673069e-01 -0.6077444258
[96,] -0.3553743506 -2.735336e-01 -0.5847825905
[97,] 0.9252331320 5.593320e-01 1.2658011561
[98,] 0.5468904170 2.597236e-01 0.4315629980
[99,] 0.4162821966 6.438214e-01 1.6171623705
[100,] 0.6382715435 -9.274246e-02 0.3655728643
[101,] 0.6970056220 6.222609e-01 0.1443412042
[102,] -0.6532657708 -3.076301e-01 0.7653202144
[103,] 0.5816515240 7.645928e-01 0.7488001787
[104,] 0.1573606968 -4.530070e-01 -0.1839519291
[105,] 1.0440545113 6.126438e-01 -0.1282912995
[106,] -1.6336580435 -1.386374e+00 -1.0094093821
[107,] -0.2734951709 -5.445205e-01 -0.8096670022
[108,] -0.9657569833 -2.502771e-01 -0.6910581170
[109,] -0.2918811600 1.619258e-01 -0.5780255139
[110,] -1.6436837879 -5.586258e-01 -1.3571895063
[111,] -0.1836800488 1.072637e-01 1.3839884187
[112,] 0.1534576614 2.830931e-01 0.5873468600
[113,] -0.8837347819 -8.946057e-01 -1.4042735538
[114,] -0.4889630892 4.974690e-01 1.4703859079
[115,] 0.0906517754 4.526356e-01 0.0346636446
[116,] 0.8041196477 -2.244867e-02 -0.0059165551
[117,] -0.8439490982 7.631856e-02 1.4338814168
[118,] 0.0236645648 1.522435e-01 0.4362889331
[119,] 0.9874183459 -7.106935e-01 -1.5205898294
[120,] -0.4791181512 2.228657e-02 0.8187945729
[121,] 0.1301040560 6.508803e-01 0.4646824643
[122,] -1.5568034454 -4.278497e-01 -0.8506471186
[123,] -0.3810199136 -1.326226e-01 0.8937790039
[124,] -0.3131398857 -6.275225e-02 0.7114673682
[125,] 0.1776352206 -4.120474e-01 0.2849700140
[126,] -1.2027420096 2.223164e-01 -0.3611823481
[127,] -0.5117572675 -6.859732e-02 0.7022913705
[128,] 0.3041619664 -1.741341e-01 -0.1180554049
[129,] 0.3552607041 3.267392e-01 0.7967877670
[130,] 0.3366141641 4.358414e-01 0.0326732865
[131,] -0.2838767187 -2.554047e-01 0.3266296487
[132,] 0.4457261398 7.830734e-01 0.2303030562
[133,] -0.3643382628 -3.799964e-01 -0.4540909693
[134,] 0.8697090896 8.865614e-01 1.5952731751
[135,] 0.1111720845 -6.792939e-01 -0.2899214623
[136,] 0.9453523696 8.211048e-01 1.5973880637
[137,] 0.8654920334 -3.716333e-01 -0.0279590242
[138,] 0.9416535662 2.577295e-01 -0.0517026728
[139,] 1.4460321200 -4.902532e-02 -0.1134510439
[140,] -0.3014256879 5.218493e-01 0.0823123828
[141,] 1.5737770602 7.838106e-01 1.7468985481
[142,] 0.0477563407 -2.712533e-01 -2.2228497168
[143,] 0.9185469841 9.185291e-01 1.8185709873
[144,] 0.9083593880 -3.799721e-01 0.2588848720
[145,] 1.4331750574 6.806470e-01 0.3598863904
[146,] 0.4460506418 6.725712e-01 0.6235579243
[147,] -2.0413370987 -1.172374e+00 -1.3765015676
[148,] 0.5221066686 -5.580021e-01 -0.2011183776
[149,] 1.1721344197 1.208608e+00 0.6243873998
[150,] 1.6151429217 9.217082e-01 1.0713216137
[151,] 0.4906882614 3.063489e-01 -1.5843726844
[152,] 0.5820796924 -8.920919e-01 0.0236302066
[153,] 0.8616151415 -3.712208e-01 0.0097644553
[154,] 0.4337669244 2.829420e-01 1.2405173324
[155,] 0.5861011686 4.851031e-01 2.0730622667
[156,] 1.0609669016 1.338238e-02 1.0761711674
[157,] 0.0748761367 -7.797860e-01 -1.5734663293
[158,] -0.0970635150 7.148284e-01 0.8811985084
[159,] 1.2475271408 1.161148e+00 0.7750707078
[160,] -1.6478353060 -1.407518e+00 -1.1853124296
[161,] -0.0590472697 -5.615106e-01 -0.4313492327
[162,] 0.5052149736 1.797925e-02 -0.0404285127
[163,] -0.2439653639 -7.779968e-01 -1.3724174462
[164,] -1.1208966167 -5.471530e-01 -0.2992734106
[165,] 0.9038561244 1.146001e-01 0.2588611256
[166,] -2.5393360161 -1.270377e+00 -1.0533800657
[167,] -0.0641192587 -1.027518e-01 0.6385547609
[168,] 0.6350964749 5.564924e-01 1.8529041171
[169,] 0.6610889498 -1.005540e-01 -0.3088240509
[170,] 0.3203489775 1.305792e-01 -0.8177542162
[171,] -2.3790622169 -1.455767e+00 -1.2987805533
[172,] 0.6057547430 -1.031096e+00 -0.1596141071
[173,] 0.1249332408 -3.686077e-01 -0.8019542839
[174,] 0.4073515769 -3.149810e-01 -0.0525967279
[175,] 0.1317920121 1.603722e-01 0.4325141154
[176,] -0.7318280889 -6.671000e-01 1.2263764852
[177,] -1.1934197227 -5.877712e-01 -0.0037250761
[178,] -1.1257262284 -1.155837e+00 -2.2788637150
[179,] -1.0427174995 -4.387680e-01 -0.7278647513
[180,] -1.6456703197 -5.191377e-01 -0.4339197691
[181,] 1.3875042454 6.230914e-01 0.6725469845
[182,] -1.1714167155 -8.236619e-01 -0.2670086823
[183,] -0.3574558163 -8.557515e-01 0.4640769929
[184,] 0.9611122251 1.182384e+00 1.2132314717
[185,] -1.3980619300 -1.633977e-01 0.4624609561
[186,] -0.0810637248 2.085422e-01 0.2649395933
[187,] -0.2309545987 -6.398164e-01 -2.2839398733
[188,] 0.2604371207 5.392775e-01 0.9245997202
[189,] -0.0424878569 -3.043987e-02 0.9204286032
[190,] 1.3636251714 4.993892e-01 0.1661919693
[191,] 0.3031921623 1.124064e-02 0.0430735382
[192,] -0.1535718517 6.815553e-02 -0.6921348085
[193,] 0.2711686302 3.809559e-01 0.9462746529
[194,] -0.5765444298 -3.671866e-01 -0.8464936378
[195,] 0.5882168805 3.492836e-01 1.5999961662
[196,] 1.3397956685 3.199422e-01 0.5476267936
[197,] -1.5155758351 -8.469226e-01 -0.8205405087
[198,] -0.0248644868 9.704456e-01 1.4334849652
[199,] -0.9595390658 1.209642e-01 0.0602869071
[200,] -0.5692293681 2.146602e-01 1.1928730758
[201,] -0.3233465634 1.750968e-01 -0.6052784497
[202,] -0.9616067025 -1.194417e+00 -1.6473411530
[203,] -1.1257317991 -9.106101e-01 -1.4600099860
[204,] 0.2058486421 6.779101e-01 0.0223759722
[205,] 1.3050316034 1.347895e+00 2.3961890391
[206,] -2.7001155260 -1.385545e+00 -1.4010252585
[207,] 0.0100674325 -5.910188e-01 -0.9645240001
[208,] -0.3779038264 -5.782807e-01 -0.3337465414
[209,] 0.4085318231 -1.375243e-01 -0.2941255124
[210,] -0.6110806234 1.779958e-01 -0.5203638337
[211,] -0.9621657212 -4.992378e-01 1.6084398633
[212,] -0.1040365188 -5.104926e-01 -0.4100897084
[213,] 1.8054686031 3.495480e-01 -1.2295529705
[214,] 0.7857727988 1.560126e+00 0.9541176718
[215,] -0.6289614544 -6.273401e-01 -0.9424790685
[216,] -1.5178551618 -5.294903e-01 -1.1774410808
[217,] 0.5463511680 -5.174450e-02 0.1027608026
[218,] 1.1515434305 5.874793e-01 0.9750453810
[219,] -0.7787040928 -7.600173e-01 -0.0184820829
[220,] -0.6049231415 -1.448061e-01 0.7864532107
[221,] 0.8067218789 1.579072e+00 1.6455197215
[222,] 0.4985619621 8.819590e-02 -0.5590375646
[223,] 2.0870378518 2.113014e-01 -0.9638976086
[224,] 0.3929218934 -2.724252e-02 0.3048857491
[225,] 0.4426600321 8.623906e-02 1.8104381139
[226,] -1.3617344451 -1.084839e+00 -1.5451010132
[227,] 1.0454605576 -1.941861e-01 -0.2608632191
[228,] 2.0010001109 5.669088e-01 0.8416040324
[229,] -0.3056821257 3.143201e-01 -0.2882059142
[230,] -0.0934537539 -1.535297e-02 -0.4219563610
[231,] 1.4669894721 1.136163e+00 -0.2239167078
[232,] 1.8832948759 1.918272e+00 2.6590568402
[233,] 1.3188883720 1.361693e-01 -0.1275364410
[234,] -0.6983739744 -3.839562e-01 -0.1953828720
[235,] -0.2019604364 4.034308e-01 0.0639746006
[236,] -0.4649983479 -1.263989e-01 0.7414992677
[237,] -0.5340753136 -1.657660e-02 -0.5745394766
[238,] 1.4491548864 2.145187e-01 0.7818117758
[239,] -0.8870622884 -3.067359e-02 -1.2821281742
[240,] -0.4572093251 3.816814e-01 0.2245678574
[241,] -0.1032690043 -4.632574e-01 -1.0129426157
[242,] -1.2067990144 -1.635450e-01 -0.7526276689
[243,] -0.3767003202 2.816157e-01 -0.2487986838
[244,] 1.7769863513 2.819761e-01 -0.2031732477
[245,] -0.4667850677 -5.001909e-01 -1.5351922385
[246,] 1.7761231128 8.013933e-01 1.7045789634
[247,] 0.2406841566 3.539307e-01 0.6208916584
[248,] 0.8171994438 9.103933e-02 -0.3650132035
[249,] -0.0413265240 8.539191e-02 0.3184061785
[250,] -0.4734319865 -7.724775e-01 -1.0705976088
[251,] -0.4815846018 2.366651e-01 0.0588741081
[252,] -1.2004035301 -4.847159e-02 0.4187528031
[253,] 0.3987438814 -2.842766e-01 -0.0785252281
[254,] 0.1943026979 5.876537e-01 -0.0746651570
[255,] 0.1194229038 -3.446876e-01 -1.2877497100
[256,] -0.2837439721 -7.396306e-01 -1.5322014625
[257,] 1.3873701648 9.396443e-01 2.8092978882
[258,] -0.1645696078 -1.426377e-02 -0.9693048906
[259,] -0.4257726342 -2.869155e-01 -0.1391358877
[260,] 0.6465667672 -2.578817e-01 0.3664923298
[261,] -0.5586616897 -6.668523e-01 -0.9141020137
[262,] -0.2431601998 -1.543633e-02 1.0297673822
[263,] 0.1897051640 -2.387356e-01 0.5298931632
[264,] -1.1844305472 -9.225178e-01 0.0614385149
[265,] -1.8333642394 -7.059100e-01 -1.1483596150
[266,] 0.0772874655 -1.344114e-01 -0.5305310869
[267,] 1.1841699389 1.560266e-01 -0.4411141458
[268,] 1.2844639463 4.070245e-01 0.8290858785
[269,] 0.1709612961 8.596084e-01 -0.0583120316
[270,] 0.3233607747 7.361074e-01 1.0829377783
[271,] -0.6792608077 3.994416e-01 0.9413627850
[272,] 0.0059636494 1.778813e-01 -0.9063842067
[273,] 0.2672142786 -1.458160e-02 1.4420588125
[274,] -3.1702995910 -1.164722e+00 -1.3739528598
[275,] 0.8372221760 -6.365684e-02 -1.1079826542
[276,] -1.1672134742 -3.394895e-01 -0.8718334751
[277,] 2.3070913251 1.023556e+00 1.6086420160
[278,] -0.5523556159 -2.679038e-01 -0.7556190462
[279,] 1.3154480835 2.137050e-01 1.1750410740
[280,] -0.1672680356 -4.786325e-01 -1.3631966277
[281,] -0.1320014291 3.567547e-01 -0.8899290578
[282,] 0.0209321163 3.324969e-01 0.5803845977
[283,] -1.1507981230 -7.811807e-01 -0.9939405282
[284,] 1.4186749779 6.920921e-02 -0.0634052590
[285,] 0.8278937434 4.119342e-01 0.0681972733
[286,] 0.4297376418 5.919978e-01 1.1906419149
[287,] -0.0487195733 5.201953e-01 1.0505431454
[288,] 0.5301297642 -6.627776e-01 -1.9938608094
[289,] -0.6676805076 2.399644e-01 -0.3507889266
[290,] -1.1866543168 -5.714432e-01 -0.4549840661
[291,] 1.3296738641 -3.037132e-01 -0.3575429937
[292,] -0.6410393068 -4.686836e-02 -0.0801738676
[293,] 0.2010760637 -5.458123e-01 -0.3404022237
[294,] 1.2560847312 9.019229e-01 1.0152358719
[295,] -1.8156411946 -7.984624e-01 -1.2244304298
[296,] -0.6087604194 -6.433610e-01 0.7505708093
[297,] -0.8521680424 -5.610586e-02 -0.6399068622
[298,] -1.2690247301 -9.354994e-01 -0.0407172545
[299,] -0.6392237540 7.474636e-01 0.2636562540
[300,] -0.3208828699 5.313820e-01 0.5924383389
[301,] -0.5917384809 -3.431409e-01 -0.9729511588
[302,] 0.7398230512 5.034846e-01 1.0292291766
[303,] -2.1315514239 -9.695901e-01 -1.1844034403
[304,] -0.9393352947 3.625334e-01 1.4738168277
[305,] -0.5078137775 -1.072228e+00 -2.1097207334
[306,] -1.5444911635 -5.978234e-01 -1.0922139277
[307,] 0.3222937015 -4.963874e-02 -0.4562186345
[308,] 0.3157290329 -6.635592e-01 -1.0246915053
[309,] 0.9671707378 7.069453e-01 0.6267841564
[310,] -1.0159860836 2.470982e-01 0.2138749006
[311,] -1.4647574613 -5.154073e-01 -0.2893393077
[312,] -0.5391129938 -8.445515e-02 0.3952272427
[313,] 0.5609670776 1.411549e-01 0.2553344909
[314,] 1.0282409697 9.088285e-01 -0.2988941169
[315,] -0.5481978141 8.139748e-02 0.5679975790
[316,] -1.0626872519 -9.435713e-01 -0.4694586464
[317,] 0.3858533066 1.031710e-01 0.5142065738
[318,] -0.7499666746 2.756825e-01 0.8157199814
[319,] 0.9798052570 4.841253e-01 1.2941888097
[320,] 0.5953488112 -4.475553e-02 -1.5532766220
[321,] -0.6372062806 -2.049611e-01 -1.8422060750
[322,] -0.7114139435 -1.309014e+00 -1.4416595011
[323,] -0.5915801616 9.704658e-02 -1.0961151771
[324,] 1.8663119070 1.096958e+00 1.1657996375
[325,] 0.0989127148 1.452490e-02 -1.0701165538
[326,] -1.7277981423 -9.409071e-01 1.4173168862
[327,] 0.5650189098 3.298338e-01 -0.0700237495
[328,] 0.1316764791 4.503867e-01 -0.7590105197
[329,] 1.2064345674 1.592536e-01 0.4163301013
[330,] -0.7306106464 -1.447086e-01 -0.5326260164
[331,] 0.9114570566 4.288888e-01 -0.5320885653
[332,] 0.7411042574 1.024646e+00 1.7370086185
[333,] 0.2742027321 -1.184494e-01 0.5345936927
[ reached getOption(“max.print”) — omitted 9667 rows ]
> n=length(Z)
> m=0 #both n and m will be used in calculating the 0.99 confidence interval for last part of Question 1a
> con.level<-0.99 # this is the level of confidence
> Zed<-sqrt(0.995) # this the z value for calculating confidence interval
> T<-Zed/sqrt(n)
> se<-sd(T) # output the standard error
> CI<-0.99*se #confidence interval
> LowerLimit<-m-CI
> UpperLimit<-m+C
> #########
> ####Question 1 B
> cat(“MLE estimate of E[Z1]=”,mean(Z[,1]),”n”)
MLE estimate of E[Z1]= 0.008552179
> cat(“MLE estimate of E[Z2]=”,mean(Z[,2]),”n”)
MLE estimate of E[Z2]= 0.0008330524
> cat(“MLE estimate of E[Z3]=”,mean(Z[,3]),”n”)
MLE estimate of E[Z3]= 0.003962192
> cat(“MLE estimate of standard deviation of Z1=”,sd(Z[,1]),”n”)
MLE estimate of standard deviation of Z1= 1.00808
> cat(“MLE estimate of standard deviation of Z2=”,sd(Z[,2]),”n”)
MLE estimate of standard deviation of Z2= 0.63917
> cat(“MLE estimate of standard deviation of Z3=”,sd(Z[,3]),”n”)
MLE estimate of standard deviation of Z3= 1.003801
> #########
> ####Question 1 C
> ##Kendall’s Correlation
> cat(“Estimate of kendall’s correlation between Z1 and Z2=”,cor(Z[,1],Z[,2],method=”kendall”), “n”)
Estimate of kendall’s correlation between Z1 and Z2= 0.4385115
> cat(“Estimate of kendall’s correlation between Z1 and Z3=”,cor(Z[,1],Z[,3],method=”kendall”), “n”)
Estimate of kendall’s correlation between Z1 and Z3= 0.2745759
> cat(“Estimate of kendall’s correlation between Z2 and Z3=”,cor(Z[,2],Z[,3]
+ ,method=”kendall”), “n”)
Estimate of kendall’s correlation between Z2 and Z3= 0.4418381
>
> ####Spearsman’s Correlation
> cat(“Estimate of Spearma’s correlation between Z1 and Z2=”,cor(Z[,1],Z[,2],method=”spearman”), “n”)
Estimate of Spearma’s correlation between Z1 and Z2= 0.6179661
> cat(“Estimate of Spearman’s correlation between Z1 and Z3=”,cor(Z[,1],Z[,3],method=”spearman”), “n”)
Estimate of Spearman’s correlation between Z1 and Z3= 0.4013849
> cat(“Estimate of Spearman’s correlation=”,cor(Z[,2],Z[,3],method=”spearman”), “n”)
Estimate of Spearman’s correlation= 0.6231296
> ####Question 1 d
> ##
> U.tcopula<-pt(Z,dof)#gennerates a sample (U1, U2) from the t- copula
> U.Gaussiancopula<-pnorm(Z,0,1) #generates sample (U1,U2) from Gaussian Copula
##2a Maximum Liklihood estimate
Xdata<-c(0.15,0.10,0.39,0.17,8.39,30.77,2.53,0.26,8.71,85.99)
Npara<-length(Xdata)
mTrue<-mean(Xdata)
SigTrue<-sd(Xdata)
sim<-exp(rnorm(1000,mTrue,SigTrue))
sigMLE<-sd(log(sim))
cat(“MLE lamda=”, “MLE sigma=”, sigMLE,”n”)
##2b Posterior Mean and sdev
#Postrior mean and standard devition
cat(“mu MCMC Posterior mean=”,mean(Xdata), “Posterior Standard deviaion=”, sd (Xdata))
## 2c
##Bayes Posterior Mean and Sdve
Posteriormean=mean(Xdata)
Posteriormean
[1] 13.746
PosteriorSdev=sd(Xdata)
PosteriorSdev
[1] 27.08933
mydata=rgamma(100,1,0.5)
nMydata=mean(mydata)
sigMydata=sd(mydata)
cat(“Alpha Estimate=”,nMydata,”Sigma Estimate=”,sigMydata)
Alpha Estimate= 1.907266 Sigma Estimate= 1.779147
Mydata
## 2a Maximum Liklihood estimate
Xdata<-c(0.15,0.10,0.39,0.17,8.39,30.77,2.53,0.26,8.71,85.99)
> Npara<-length(Xdata)
> mTrue<-mean(Xdata)
> SigTrue<-sd(Xdata)
> sim<-exp(rnorm(1000,mTrue,SigTrue))
> sigMLE<-sd(log(sim))
> cat(“MLE lamda=”, “MLE sigma=”, sigMLE,”n”)
MLE lamda= MLE sigma= 27.16486
##2b Posterior Mean and sdev
|
> #Postrior mean and standard devition > cat(“mu MCMC Posterior mean=”,mean(Xdata), “Posterior Standard deviaion=”, sd (Xdata)) mu MCMC Posterior mean= 13.746 Posterior Standard deviaion= 27.08933
|
|
|
|
|
|
###Question 2d
mydata=rgamma(100,1,0.5)
> nMydata=mean(mydata)
> sigMydata=sd(mydata)
> cat(“Alpha Estimate=”,nMydata,”Sigma Estimate=”,sigMydata)
Alpha Estimate= 1.907266 Sigma Estimate= 1.779147
> Mydata
[1] 2.44136898 0.85149048 0.41384965 0.84938249 0.58331153 1.27044932 2.12686491
[8] 1.36902755 9.52569191 5.68138784 0.76164651 1.92839205 0.67777985 2.06393150
[15] 0.72339954 3.62025169 0.17015991 0.63162312 0.73862618 6.11082158 0.34913997
[22] 4.95750348 6.29234613 8.63575314 2.70652608 1.01832155 0.58920045 2.30464240
[29] 1.28160445 1.47862286 1.34225547 0.21040981 0.43350751 0.61302264 2.82009340
[36] 0.67102777 0.05862084 3.90269521 2.02580475 2.72265960 1.21655099 0.43253271
[43] 0.45952903 0.47923713 0.02389339 0.89271301 2.84006227 4.54072762 1.46763958
[50] 2.14184923 0.81090395 2.29649594 0.51929692 3.38478321 0.15130158 1.13247705
[57] 1.01459582 3.19501380 0.09954389 0.77016261 0.55715698 1.48989538 0.58341984
[64] 0.47705564 0.15832176 1.75816313 4.04534005 0.27093537 0.76474419 0.60988719
[71] 0.17588551 0.34920563 0.28925759 0.79391739 2.48770786 0.04930602 2.59389880
[78] 2.86008462 4.52476874 0.84937332 0.39871474 0.78114947 0.44275953 0.64414873
[85] 3.99213029 3.16239544 2.40686075 7.15048162 0.40343135 0.87551093 0.91287086
[92] 0.05226923 2.54082561 1.85557218 0.75560271 1.49962976 1.79272573 1.02665746
[99] 1.01812117 2.24809242
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