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ACST840 Quantitative Research Methods For Operational Risk Modelling

Questions:

Assume that the corresponding annual number of loss events Nt , t = 1, ..., 4 are independent and identically distributed from P oisson(λ) and severities Xi are independent and identically distributed from lognormal distribution LN (µ, σ2). Also Nt and Xi are all independent.
a) Estimate λ, µ, σ using the maximum likelihood method.
b) Using Markov chain Monte Carlo method (random walk Metropolis-Hastings within Gibbs
algorithm studied during the course) with 104
chain iterations, under the Bayesian approach
with constant priors, estimate the posterior mean and the posterior standard deviation for
λ, µ, σ.
c) Using closed form solution, under the Bayesian approach with constant priors, estimate the
posterior mean and posterior standard deviation for λ.
d) Under the Bayesian approach, model parameter λ is modelled by random variable Λ. Suppose that expert’s best estimates for the expected number of annual losses and associated
uncertainty are E[Λ] = 1 and Vco[Λ] = 0.5. If the prior distribution for Λ is Gamma(α, β),
find α and β using expert estimates and calculate (using closed form solution and MCMC)
the posterior mean and posterior standard deviation for Λ.

Answer:

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&


lt;-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

 

 

>

 

 

> ## 2c

> ##Bayes Posterior Mean and Sdve

> Posteriormean=mean(Xdata)

> Posteriormean

[1] 13.746

> PosteriorSdev=sd(Xdata)

> PosteriorSdev

[1] 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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