Aggregating a series of correlated lognormal variables

Generating observations and log-normally distributed random errors

We generate 10000 Observations with mean 10 and multiplicative standard deviation of 1.5.

nObs <- 200; nRep <- 1000
#nObs <- 1000; nRep <- 100
xTrue <- rep(10, nObs)
sigmaStar <- rep(1.5, nObs) # multiplicative stddev of 1.2
theta <- getParmsLognormForExpval(xTrue, sigmaStar)
# generate observations with correlated errors
acf1 <- c(0.4,0.1)
corrM <- setMatrixOffDiagonals(
  diag(nrow = nObs), value = acf1, isSymmetric = TRUE)
xObsN <- exp(mvtnorm::rmvnorm(
  nRep, mean = theta[,1]
  , sigma = diag(theta[,2]) %*% corrM %*% diag(theta[,2])))
ds <- tibble(i = 1:nObs, xTrue, xObs = xObsN[1,], xErr =  xObs - xTrue)
summary(rowSums(xObsN))

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1728    1941    2000    2000    2053    2353

#plot(density(rowSums(xObsN)))

Estimating the correlation matrix and effective number of parameters

  (effAcf <- computeEffectiveAutoCorr(ds$xErr))

## [1] 1.0000000 0.2078811

  (nEff <- computeEffectiveNumObs(ds$xErr))

## [1] 141.4744

Due to autocorrelation, the effective number of parameters is less than nObs = R nObs.

Computing the mean and its standard deviation

coefSum <- estimateSumLognormal( theta[,1], theta[,2], effAcf = effAcf )
(sumExp <- getLognormMoments( coefSum[1], coefSum[2])[1,"mean"])

## mean 
## 2000