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Bayesian Estimation for Bivariate Gamma Processes with Copula

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Bayesian Inference and Computation in Reliability and Survival Analysis

Part of the book series: Emerging Topics in Statistics and Biostatistics ((ETSB))

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Abstract

Gamma stochastic process has been proposed to replace Brownian motion and geometric Brownian motion to characterize the degradation measurements from an accelerated degradation test (ADT). Tsai et al. [27] applied bivariate Gamma process to model the two-variable ADT under the independent assumption. In this chapter, we consider three bivariate Gamma processes utilizing the Clayton, Frank, and Gumbel copulas to describe the dependence characteristics of bivariate Gamma variables. Owing to the complex structure of modeling, the differentiation-based likelihood estimation of the copulas are not always tractable. Bayesian analysis using Markov chain Monte Carlo method is an effective alternative to implement parameter estimation and model comparisons. Extensive simulation studies that calculate the mean square errors (MSEs) of the derived estimates are conduced to show the efficiency of the proposed method. Three data sets from the Clayton, Frank, and Gumbel copulas are analyzed and used for the demonstration of model selections via Bayesian approach.

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Author information

Authors and Affiliations

  1. Applied Mathematics Department, Chung Yuan Christian University, Taoyuan City, Taiwan

    Yu-Jau Lin

  2. Department of Statistics, Tamkang University, New Taipei City, Taiwan

    Tzong-Ru Tsai

  3. Department of Mathematical Sciences, University of South Dakota, Vermillion, SD, USA

    Yuhlong Lio

Authors
  1. Yu-Jau Lin
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  2. Tzong-Ru Tsai
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  3. Yuhlong Lio
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Corresponding author

Correspondence to Tzong-Ru Tsai .

Editor information

Editors and Affiliations

  1. Department of Mathematical Sciences, University of South Dakota, Vermillion, SD, USA

    Yuhlong Lio

  2. College of Health Solutions, Arizona State University, Phoenix, AZ, USA

    Ding-Geng Chen

  3. Department of Mathematical Sciences, Bentley University, Waltham, MA, USA

    Hon Keung Tony Ng

  4. Department of Statistics, Tamkang University, New Taipei City, Taiwan

    Tzong-Ru Tsai

Appendix

Appendix

Below are the R codes via NIMBLE package in Bayesian using Gibbs sampling (BUGS) language to estimate the parameters (s1, r2, s2, r2), including copula parameter (theta).

# install.packages("copula")         ## R copula package                                         (r-nimble.org) # install.packages("nimble", type="source") #    NIMBLE needs a C++ compiler and the standard utility 'make'      to generate and compile C++ library(nimble)                  ## call NIMBLE package library(copula)                  ## call copula package                                  ## Generate bivariate Gamma                                  variates from  Clayton copula theta =3 # the target copula parameter n1=n=3                           ## no. of units used for each                                     treatment C=50 k=6                              ## set up k=6 treatments tU=168∗26                        ## 26 weeks (one week=168                                     hours) for ADT t=seq(0,tU,168∗2)/24             ## check the test each two weeks m=length(t)-1                    ## no. of measurement times dt=t[2:(m+1)]-t[1:m]             ## scaled time increments LL=L=array(dim=c(6,2)) LL[1,]=c(25, 350) LL[2,]=c(45, 650) LL[3,]=c(60, 650) LL[4,]=c(75, 450) LL[5,]=c(75, 550) LL[6,]=c(75, 650) mL1=max(LL[,1]) mL2=max(LL[,2]) L[,1]=(1/(25+273.15)-1/(LL[,1]+273.15))/(1/(25+273.15)-1/ (mL1+273.15)) L[,2]=(log(LL[,2])-log(350))/(log(mL2)-log(350)) r0=-2.902;  r1=0.577; r2=0.533; r3=0.531; be=0.662 v.r=c(r0,r1,r2,r3,be) generatedata=function(v,t,be,n1, theta)  {m=length(t)-1           # no. of measurement times   y=array(dim=c(2,m,n1,k)) for (i in 1:k) {for (j in 1:n1){      cop <- claytonCopula(param = theta, dim = 2)      U <- rCopula(m, cop)      y[1,,j,i]= qgamma(U[,1],shape=v[i]∗dt[i],scale=be)      y[2,,j,i]= qgamma(U[,2],shape=v[i]∗dt[i],scale=be)      } } return(y) # 4 dimensional data } v=rep(0,k)                                                                                                                            # the drift parameter for each treatment v0=exp(r0) for (i in 1:k) {v[i]=v0∗exp(r1∗L[i,1]+r2∗L[i,2]+r3∗L[i,1]                      ∗L[i,2])} #delta yy= generatedata (v,t,be,n1, theta)  # generate the bivariate                                        copula data y1=yy[1,,,]; y2=yy[2,,,] zeros=array(0,dim=c(m,n1,k)) mydata=list(xx=y1, yy=y2,  L=L , r3=0.531) # the data set myinits=function()list(theta=5, r0=-2.8,  r1=0.4, r2=0.4,                         be=0.5) # initial values ClaytonCode=nimbleCode({           # likelihood defined below for (mm in 1:13){ for( nn in 1:n1){   for (ii in 1:6){     delta[mm,nn,ii] <- (exp(r0)∗exp(r1∗L[ii,1]+r2∗L[ii,2]+r3                                     ∗L[ii,1]∗L[ii,2]))∗ 14     dummy[mm,nn,ii] ~ dpois(negLogLike[mm,nn,ii]) # zero trick     u[mm,nn,ii]<- pgamma(xx[mm,nn,ii], shape=delta[mm,nn,ii],                       scale=be)      # marginal gamma CDF     v[mm,nn,ii]<- pgamma(yy[mm,nn,ii], shape=delta[mm,nn,ii],                                        scale=be)     negLogLike[mm,nn,ii]<-  100+(-1)∗logLike[mm,nn,ii]     logLike[mm,nn,ii]<- log(theta+1)+(-theta-1)∗log(u[mm,nn,ii]                                                  ∗v[mm,nn,ii])+           ((-1)∗(2∗theta+1)/theta)∗log(u[mm,nn,ii]ˆ(-theta)+           v[mm,nn,ii]ˆ(-theta)-1)+           log(dgamma(xx[mm,nn,ii], delta[mm,nn,ii],scale=be))+           log(dgamma(yy[mm,nn,ii],delta[mm,nn,ii],scale=be))   ## negLogLike[mm,nn,ii]= (-1)∗log-likelihood, which is      "the negative value of the log-likelihood function"   ## logLike[mm,nn,ii] is the log-likelihood of Eq.(3) }}} theta~ dgamma(.01,.01)    # parameter’s prior distributions r0  ~ dunif(-3.5,-2.5) r1  ~ dgamma(.01,.01) r2  ~ dgamma(.01,.01) be ~  dgamma(.01,.01) }) # end of nimbleCode function                                              # computing the                                              estimation by MCMC                                              method mcmc.out=nimbleMCMC(ClaytonCode,             # call  "Clayton                                              Code" nimbleCode  monitors=c("theta","r0", "r1", "r2","be"),  # interested param-                                                  eters  constants=list(xx=y1, yy=y2, L=L, r3=0.531, n1=n1),  data=list(dummy=zeros),                     # data and constants  inits=myinits, thin=1,                      #  initial values  nchains=2, niter=20000, nburnin=1000,       # 2 chains, 20000                                                MCMC iterations,                                                1000 burn-in periods  summary=TRUE)                               #  summary of the MCMC                                                estimation mcmc.out$summary                             # the MCMC estimates ## The Bayes estimates:  be=0.6242, r0=-2.8598, r1= 0.4528,    r2=0.6806, theta= 2.7733 ## Below are the sample MCMC outputs. #$chain1 #             Mean    Median  St. Dev. 95%CI_low 95%CI_upp #    be      0.6267   0.6231   0.0619    0.5205        0.7647 #    r0     -2.8652  -2.8627   0.1334   -3.1288       -2.6053 #    r1      0.4543   0.4579   0.1364    0.1846        0.7146 #    r2      0.6802   0.6780   0.1255    0.4274        0.9284 #    theta  2.7884   2.7655   0.3917    2.0927        3.6011 #$chain2 #            Mean    Median  St. Dev.  95%CI_low 95%CI_upp #    be      0.6217   0.6183   0.0629    0.5083        0.7606 #    r0     -2.8545  -2.8513   0.1357   -3.1315       -2.6027 #    r1      0.4513   0.4509   0.1303    0.1917        0.7012 #    r2      0.6810   0.6815   0.1218    0.4339        0.9195 #    theta  2.7581   2.7353   0.3877    2.0658        3.5689 #$all.chains #            Mean    Median  St. Dev. 95%CI_low 95%CI_upp #    be      0.6242   0.6205   0.0625    0.5141        0.7623 #    r0     -2.8598  -2.8568   0.1347   -3.1300       -2.6035 #    r1      0.4528   0.4544   0.1334    0.1886        0.7081 #    r2      0.6806   0.6796   0.1237    0.4306        0.9247 #    theta  2.7733   2.7475   0.3900    2.0775        3.5940

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Lin, YJ., Tsai, TR., Lio, Y. (2022). Bayesian Estimation for Bivariate Gamma Processes with Copula. In: Lio, Y., Chen, DG., Ng, H.K.T., Tsai, TR. (eds) Bayesian Inference and Computation in Reliability and Survival Analysis. Emerging Topics in Statistics and Biostatistics . Springer, Cham. https://doi.org/10.1007/978-3-030-88658-5_8

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