# Does a median-unbiased estimator minimize mean absolute deviance?

This is a follow-up but also a different question of my previous one.

I read on Wikipedia that "A median-unbiased estimator minimizes the risk with respect to the absolute-deviation loss function, as observed by Laplace." However, my Monte Carlo simulation results does not support this argument.

I assume a sample from a log-normal population, $X_1,X_2,...,X_N \sim \mbox{LN}(\mu,\sigma^2)$,where, $\mu$ and $\sigma$ are the log-mean and log-sd,$\beta = \exp(\mu)=50$

The geometric-mean estimator is a median-unbiased estimator for the population median $\exp(\mu)$,

$\hat{\beta}_{\mbox{GM}}= \exp(\hat{\mu})= \exp{(\sum\frac{\log(X_i)}{N})} \sim \mbox{LN}(\mu,\sigma^2/N)$ where, $\mu$ and $\sigma$ are the log-mean and log-sd, $\hat\mu$ and $\hat\sigma$ are the MLEs for $\mu$ and $\sigma$.

While a corrected geometric-mean estimator is a mean-unbiased estimator for the population median.

$\hat{\beta}_{\mbox{CG}}= \exp(\hat{\mu}-\hat\sigma^2/2N)$

I generate samples of size 5 repeatedly from the LN$(\log(50),\sqrt{\log(1+2^2)})$. The replication number is 10,000. The average absolute deviations I got are 25.14 for the geometric-mean estimator and 22.92 for corrected geometric-mean. Why?

BTW, the estimated median absolute deviations are 18.18 for geometric-mean and 18.58 for corrected geometric-mean estimator.

The R script I used are here:

#{r stackexchange}
#' Calculate the geomean to estimate the lognormal median.
#'
#' This function Calculate the geomean to estimate the lognormal
#' median.
#'
#' @param x a vector.
require(plyr)
GM <- function(x){
exp(mean(log(x)))
}
#' Calculate the bias corrected geomean to estimate the lognormal
#' median.
#'
#' This function Calculate the bias corrected geomean using the
#' variance of the log of the samples, i.e., $\hat\sigma^2=1/(n-1) # \Sigma_i(\Log(X_i)-\hat\mu)^2$
#'
#' @param x a vector.
BCGM <- function(x){
y <- log(x)
exp(mean(y)-var(y)/(2*length(y)))
}
#' Calculate the bias corrected geomean to estimate the lognormal
#' median.
#'
#' This function Calculate the bias corrected geomean using
#' $\hat\sigma^2=1/(n)\Sigma_i(\Log(X_i)-\hat\mu)^2$
#'
#' @param x a vector.
CG <- function(x){
y <- log(x)
exp(mean(y)-var(y)/(2*length(y))*(length(y)-1)/length(y))
}

############################

simln <- function(n,mu,sigma,CI=FALSE)
{
X <- rlnorm(n,mu,sigma)
Y <- 1/X
gm <- GM(X)
cg <- CG(X)
##gmk <- log(2)/GM(log(2)*Y) #the same as GM(X)
##cgk <- log(2)/CG(log(2)*Y)
cgk <- 1/CG(Y)
sm <- median(X)
if(CI==TRUE) ci <- calCI(X)
##bcgm <- BCGM(X)
##return(c(gm,cg,bcgm))
if(CI==FALSE) return(c(GM=gm,CG=cg,CGK=cgk,SM=sm)) else return(c(GM=gm,CG=cg,CGK=cgk,CI=ci,SM=sm))
}
cv <-2
mcN <-10000
res <- sapply(1:mcN,function(i){simln(n=5,mu=log(50),sigma=sqrt(log(1+cv^2)), CI=FALSE)})
sumres.mse <- apply(res,1,function(x) mean((x-50)^2))
#

#{r eval=FALSE}
GM       CG      CGK       SM
#25.14202 22.91564 29.65724 31.49275
#> sumres.mse
GM       CG      CGK       SM
#1368.209 1031.478 2051.540 2407.218
#

• 1.) "10,000" is too small for your question--try "250,000" (or more). 2.) If you run a Monte Carlo simulation and get a result that seems strange, try changing the seed with set.seed. 3.) Don't always trust Wikipedia--note how your quoted text (from the "Median" article) differs from this other Wikipedia article 4.) Your R code is a total mess--check out Google's R Style Guide for some good style guidelines. – Steve S Jul 7 '14 at 13:34

If we choose an estimator $\alpha^+$ by the criterion that it minimizes the expected absolute error from the true value $\alpha$

$E=<|\alpha^+-\alpha|> = \int_{-\infty}^{\alpha^+} (\alpha^+-\alpha)f(\alpha) \mathrm{d}\alpha + \int^{\infty}_{\alpha^+} (\alpha-\alpha^+)f(\alpha)\mathrm{d}\alpha$

we require

$\frac{dE}{d\alpha^+} = \int_{-\infty}^{\alpha^+} f(\alpha) \mathrm{d}\alpha - \int^{\infty}_{\alpha^+} f(\alpha) \mathrm{d}\alpha = 0$

which is equivalent to $P(\alpha > \alpha^+) = 1/2$. So $\alpha^+$ is the shown to be the median as following Laplace in 1774.

If you are having trouble with R please ask it in another question on Stack Overflow

• Theoretically, I think it is correct. However, I am confused by the R simulation results which does not back up this statement as expected. – Zhenglei Jul 15 '14 at 9:47
• I am a Data Scientist/Physicist so have never seen a line of R. As I suggested in the question, if it is a code issue you should ask it in Stack Overflow and you will get much more attention. However, the above answer is correct unless you would like to elaborate on how it generalizes to a median-unbiased estimator. For more details see page 172 of E.T. Jaynes book Probability theory ISBN 978-0-521-59271-0. – Keith Jul 15 '14 at 10:25
• Thank you a lot for your answer. It is not a coding issue. I just want to do simulations to show that a median-unbiased estimator will minimize the expected absolute deviation. I haven't accepted the answer because I am mainly confused about the simulation step. I implemented it in R but simulations could be done in Matlab or Python or any other languages. – Zhenglei Jul 15 '14 at 12:14
• I suspect the issue is that you are dealing with an approximation which works as N -> $\infty$ but you have 10,000 and 5 whcih are both small numbers. Perhaps you are better off asking three questions. Why it is true in theory, when is N practically large enough and if there is something wrong with your R code. I answered the first, the second is largely calculational but there may be a good rule of thumb for this specific case and the third belongs on stack overflow. – Keith Jul 15 '14 at 12:35
• @Keith sorry for my weak math, but can you show more detail on how you derived the expectation? – AdamO Dec 18 '17 at 17:57