
## EM for the estimate the covariance matrix

## function to calculate the parameters
par.est<-function(pM, pS){
  sX2<-pS[1,1]
  sM2<-pS[2,2]
  sY2<-pS[3,3]
  sXM<-pS[1,2]
  sXY<-pS[1,3]
  sMY<-pS[2,3]
  mX <-pM[1]
  mM <-pM[2]
  mY <-pM[3]

  pa<-sXM/sX2
  pb<-(sMY*sX2-sXM*sXY)/(sX2*sM2-sXM^2)
  pc<-(sXY*sM2-sXM*sMY)/(sX2*sM2-sXM^2)
  ps1<-sX2
  ps2<-sM2-sXM^2/sX2
  ps3<-(sX2*sM2*sY2-sX2*sMY^2-sM2*sXY^2-sY2*sXM^2+2*sXM*sXY*sMY)/(sX2*sM2-sXM^2)

  pm1<-mX
  pm2<-mM - pa*mX
  pm3<-mY - pb*mM - pc*mX

  return(c(pm2,pm3,pa,pb,pc,ps2,ps3))
}

## the M-step
M_step<-function(sM, sS, N){
## sM: first moment vector
## sS: second moment

  pM <- sM/N

  pS <- sS/(N-1) - N/(N-1)*pM%*%t(pM)
  

  return(list(pM=pM, pS=pS))

}

## the E-step
E_step<-function(dset, pM, pS){

X<-dset[,1]
M<-dset[,2]
Y<-dset[,3]

n<-dim(dset)[1]

  sX2<-rep(0,n)
  sM2<-rep(0,n)
  sY2<-rep(0,n)
  sXM<-rep(0,n)
  sXY<-rep(0,n)
  sMY<-rep(0,n)
  mX <-rep(0,n)
  mM <-rep(0,n)
  mY <-rep(0,n)

for (i in 1:n){

   ## process missing data in X
   if (!is.na(X[i])){
      ## X is available
      if (!is.na(M[i])){
         ## M is available
         if (!is.na(Y[i])){
            ## X, M, Y is available
            mX[i]<-X[i]
            mM[i]<-M[i]
            mY[i]<-Y[i]
            sX2[i]<-X[i]^2
            sY2[i]<-Y[i]^2
            sM2[i]<-M[i]^2
            sXM[i]<-X[i]*M[i]
            sXY[i]<-X[i]*Y[i]
            sMY[i]<-M[i]*Y[i]
            
         }else{
            ## X & M observed but Y is missing
            mX[i]<-X[i]
            mM[i]<-M[i]
            mY[i]<-pM[3] + pS[3,1:2]%*%solve(pS[1:2,1:2])%*%(c(X[i]-pM[1], M[i]-pM[2]))
            sX2[i]<-mX[i]*mX[i]
            sM2[i]<-mM[i]*mM[i]
            sY2[i]<-mY[i]*mY[i]+pS[3,3]-pS[3,1:2]%*%solve(pS[1:2,1:2])%*%(pS[1:2,3])
            sXM[i]<-mX[i]*mM[i]
            sXY[i]<-mX[i]*mY[i]
            sMY[i]<-mM[i]*mY[i]
         }
      }else{
         ## M is missing
         if (!is.na(Y[i])){
            ## X, Y is available but M is missing
            mX[i]<-X[i]
            mM[i]<-pM[2] + pS[2,c(1,3)]%*%solve(pS[c(1,3),c(1,3)])%*%(c(X[i]-pM[1],Y[i]-pM[3]))
            mY[i]<-Y[i]
            sX2[i]<-mX[i]*mX[i]
            sM2[i]<-mM[i]*mM[i]+pS[2,2]-pS[2,c(1,3)]%*%solve(pS[c(1,3),c(1,3)])%*%(pS[c(1,3),2])
            sY2[i]<-mY[i]*mY[i]
            sXM[i]<-mX[i]*mM[i]
            sXY[i]<-mX[i]*mY[i]
            sMY[i]<-mM[i]*mY[i]
            
         }else{
            ## X is observed but M and Y are missing
            mX[i]<-X[i]
            mM[i]<-pM[2] + pS[1,2]/pS[1,1]*(X[i]-pM[1])
            mY[i]<-pM[3] + pS[1,3]/pS[1,1]*(X[i]-pM[1])
            sX2[i]<-mX[i]*mX[i]
            sM2[i]<-mM[i]*mM[i] + pS[2,2]-pS[1,2]^2/pS[1,1]
            sY2[i]<-mY[i]*mY[i] + pS[3,3]-pS[1,3]^2/pS[1,1]
            sXM[i]<-mX[i]*mM[i]
            sXY[i]<-mX[i]*mY[i]
            sMY[i]<-mM[i]*mY[i] + pS[2,3]-pS[1,2]*pS[1,3]/pS[1,1] 
         }
      }
   }else{
      ## X is missing
      if (!is.na(M[i])){
         ## M is available
         if (!is.na(Y[i])){
            ## X is missing, M & Y are available
            mX[i]<-pM[1] + pS[1,2:3]%*%solve(pS[2:3,2:3])%*%(c(M[i]-pM[2], Y[i]-pM[3]))
            mM[i]<-M[i]
            mY[i]<-Y[i]
            sX2[i]<-mX[i]*mX[i]+pS[1,1]-pS[1,2:3]%*%solve(pS[2:3,2:3])%*%pS[2:3,1]
            sM2[i]<-mM[i]*mM[i]
            sY2[i]<-mY[i]*mY[i]
            sXM[i]<-mX[i]*mM[i]
            sXY[i]<-mX[i]*mY[i]
            sMY[i]<-mM[i]*mY[i]
         }else{
            ## Y and X is missing but M is available
            mX[i]<-pM[1]+pS[1,2]/pS[2,2]*(M[i]-pM[2])
            mM[i]<-M[i]
            mY[i]<-pM[3]+pS[2,3]/pS[2,2]*(M[i]-pM[2])
            sX2[i]<-mX[i]*mX[i]+pS[1,1]-pS[1,2]^2/pS[2,2]
            sM2[i]<-mM[i]*mM[i]
            sY2[i]<-mY[i]*mY[i]+pS[3,3]-pS[2,3]^2/pS[2,2]
            sXM[i]<-mX[i]*mM[i]
            sXY[i]<-mX[i]*mY[i]+pS[1,3]-pS[1,2]*pS[2,3]/pS[2,2]
            sMY[i]<-mM[i]*mY[i]            
         }
      }else{
         ## X, M is missing
            mX[i]<-pM[1]+pS[1,3]/pS[3,3]*(Y[i]-pM[3])
            mM[i]<-pM[2]+pS[2,3]/pS[3,3]*(Y[i]-pM[3])
            mY[i]<-Y[i]
            sX2[i]<-mX[i]*mX[i]+pS[1,1]-pS[1,3]^2/pS[3,3]
            sM2[i]<-mM[i]*mM[i]+pS[2,2]-pS[2,3]^2/pS[3,3]
            sY2[i]<-mY[i]*mY[i]
            sXM[i]<-mX[i]*mM[i]+pS[1,2]-pS[1,3]*pS[2,3]/pS[3,3]
            sXY[i]<-mX[i]*mY[i]
            sMY[i]<-mM[i]*mY[i]
      }

   }
}
     sM<-c(sum(mX),sum(mM),sum(mY))
     sS<-array(c( sum(sX2),sum(sXM),sum(sXY),sum(sXM),sum(sM2),sum(sMY),sum(sXY),sum(sMY),sum(sY2)),dim=c(3,3))
     return(list(sM=sM, sS=sS))
}

## bootstrap data using the stratified methods

bootdata<-function(dset){
  N<-dim(dset)[1]
  pat1<-NULL
  pat2<-NULL
  pat3<-NULL
  pat4<-NULL
  pat5<-NULL
  pat6<-NULL
  pat7<-NULL

  n1<-0
  n2<-0
  n3<-0
  n4<-0
  n5<-0
  n6<-0
  n7<-0

## get the number of patterns of the missing data
  for (i in 1:N){
     if (!is.na(dset[i,1])){
        if (!is.na(dset[i,2])){
           if (!is.na(dset[i,3])){
              pat1<-rbind(pat1,dset[i,])
              n1<-n1+1
           }else{
              pat2<-rbind(pat2,dset[i,])
              n2<-n2+1
           }
        }else{
           if (!is.na(dset[i,3])){
              pat3<-rbind(pat3,dset[i,])
              n3<-n3+1
           }else{
              pat4<-rbind(pat4,dset[i,])
              n4<-n4+1
           }
        }
     }else{
        if (!is.na(dset[i,2])){
           if (!is.na(dset[i,3])){
              pat5<-rbind(pat5,dset[i,])
              n5<-n5+1
           }else{
              pat6<-rbind(pat6,dset[i,])
              n6<-n6+1
           }
        }else{
           if (!is.na(dset[i,3])){
              pat7<-rbind(pat7,dset[i,])
              n7<-n7+1
           }
        }
     
     }
  }


## resample the data
  if (n1>0) pat1<-pat1[sample(1:n1, replace=T),]
  if (n2>0) pat2<-pat2[sample(1:n2, replace=T),]
  if (n3>0) pat3<-pat3[sample(1:n3, replace=T),]
  if (n4>0) pat4<-pat4[sample(1:n4, replace=T),]
  if (n5>0) pat5<-pat5[sample(1:n5, replace=T),]
  if (n6>0) pat6<-pat6[sample(1:n6, replace=T),]
  if (n7>0) pat7<-pat7[sample(1:n7, replace=T),]

  temp<-rbind(pat1,pat2,pat3,pat4,pat5,pat6,pat7)

  return(temp)  
}


## Function to run EM, listwise delete, and pairwise delete just once
runonce<-function(compdset, dset,N){
  ##complete data case
  comp<-par.est(apply(compdset,2,mean), cov(compdset))
  ## the list wise delete method
  isna<-is.na(dset)
  sumisna<-apply(isna,1,sum)
  listdata<-dset[sumisna==0,]

  temp<-cov(listdata,use='complete.obs')
  listdel<-par.est(c(mean(listdata[,1]),mean(listdata[,2]),mean(listdata[,3])),temp)

  ## pairwise delete
  temp<-cov(dset,use='pairwise.complete.obs')
  pairdel<-par.est(c(mean(dset[,1],na.rm=T),mean(dset[,2],na.rm=T),mean(dset[,3],na.rm=T)),temp)

  ## EM algorithm
  para$pM<-c(mean(listdata[,1]),mean(listdata[,2]),mean(listdata[,3]))
  para$pS<-temp
  e<-1
  while (e>.0000001){
     E.res<-E_step(dset,para$pM, para$pS)
     M.res<-M_step(E.res$sM, E.res$sS, N)
     e<-sum(abs(M.res$pM-para$pM))+sum(abs(M.res$pS-para$pS))
     para<-M.res
  }
  em<-par.est(para$pM,para$pS)
  
  results<-c(comp, comp[3]*comp[4], listdel, listdel[3]*listdel[4], pairdel, pairdel[3]*pairdel[4], em, em[3]*em[4]) 
  return(results)
}

## jackknife to get the accelerated factor a
jack<-function(dset){
  N<-dim(dset)[1]
  index<-1:N
  jackres<-array(NA, dim=c(N,8))
  for (i in 1:N){
    jackdata<-dset[index[-i],]
    para$pM<-c(mean(jackdata[,1],na.rm=T),mean(jackdata[,2],na.rm=T),mean(jackdata[,3],na.rm=T))
    para$pS<-cov(jackdata,use='complete.obs')

    e<-1
    while (e>.0000001){
      E.res<-E_step(jackdata,para$pM, para$pS)
      M.res<-M_step(E.res$sM, E.res$sS, N)
      e<-sum(abs(M.res$pM-para$pM))+sum(abs(M.res$pS-para$pS))
      para<-M.res
    }
    tempres<-par.est(para$pM,para$pS)
    jackres[i,]<-c(tempres, tempres[3]*tempres[4])
  }

  a<-rep(0,8)
  for (j in 1:8){
    mu<-mean(jackres[,j])
    a[j]<-1/6*sum((mu-jackres[,j])^3)/((sum((mu-jackres)^2))^3/2)
  }
  return(a)
}

## Function to run the bootstrap method once
bootonce<-function(compdset, dset, B, N, alpha){
  ## run original data to get thetahat
  thetahat<-runonce(compdset, dset,N)
  
  results<-array(NA, dim=c(B,32))
  for (b in 1:B){
    bcomp<-compdset[sample(1:N, replace=T),]
    bdata<-bootdata(dset)
    results[b,]<-runonce(bcomp, bdata, N)
  }
  
  ## calculate the confidence interval
  alpha<-c(.025, .05, .95, .975)
  allCI<-NULL
  for (i in 1:32){
    z0<-qnorm(sum(results[,i]