Geometric mean ≈ MLE of median [Regulatives / Guidelines]

posted by Helmut Homepage – Vienna, Austria, 2024-11-14 10:54 (217 d 17:47 ago) – Posting: # 24275
Views: 11,026

Hi BEQool,

❝ ❝ Well, the geometric mean is the maximum likelihood estimator of the median.

❝ Is geometric mean always equal to median when we have "perfect" log-normal distribution of data set (very large sample)?

Yes. Try a simple simulation:

set.seed(20241114)
mue  <- 1
CV   <- 0.2
n    <- 1e8
x    <- rlnorm(n = n, meanlog = log(mue), sdlog = sqrt(log(CV^2 + 1)))
# instead of prod(x)^(1/n) we use exp(mean(log(x))) because with
# very large n we will underrun the numeric resolution of the machine

y    <- data.frame(estimator = c("arithmetic mean", "geometric mean", "median"),
                   estimate  = c(mean(x), exp(mean(log(x))), median(x)))
y$RE <- sprintf("%+.5f%%", 100 * (y$estimate - mue) / mue)
print(y, row.names = FALSE)

       estimator estimate        RE
 arithmetic mean 1.019806 +1.98057%
  geometric mean 1.000005 +0.00046%
          median 1.000012 +0.00115%


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