Operational Risk Modeling Assignment: T-Copula and R Code Analysis

This assignment delves into operational risk modeling using R code. It begins with a Monte Carlo simulation of a Bivariate T-Copula, setting a seed and defining degrees of freedom and correlation parameters. The code generates and analyzes three risk factors, calculating expected values, variance, and confidence intervals. It then estimates the maximum likelihood of the mean and standard deviation of the risk factors. Furthermore, the assignment calculates Kendall's and Spearman's correlations between the variables to measure the strength of association. The R code generates samples from the t-copula and a Gaussian copula. The output includes results from the R code, providing estimates and correlations. The assignment covers key concepts in financial risk management, providing a comprehensive analysis of operational risk using statistical methods and simulation techniques. The document includes the R code and the corresponding outputs for each step of the analysis.

Operational Risk- Modelling
Institution:
Student Name:
Contents
QUESTION 1 R- CODE.............................................................................................................................3

QUESTION 1 OUTPUT............................................................................................................................4
QUESTION 2 R CODE............................................................................................................................12
QUESTION 2 OUTPUT..........................................................................................................................13
References...........................................................................................................................................15
QUESTION 1 R- CODE
####Question 1 A

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
QUESTION 1 OUTPUT
####Question 1 A
> 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
>
>
QUESTION 2 R CODE
##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