Bioequivalence and Bioavailability Forum • Geometric mean ≈ MLE of median

Geometric mean ≈ MLE of median [Regulatives / Guidelines]

posted by Helmut – Vienna, Austria, 2024-11-14 10:54 (190 d 18:43 ago) – Posting: # 24275
Views: 5,523

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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