the random variable that was always less than its mean…

Although this is far from a paradox when realising why the phenomenon occurred, it took me a few lines to understand why the empirical average of a log-normal sample is apparently a biased estimator of its mean. And why the biased plug-in estimator does not appear to present a bias. The picture below compares two estimators of the mean of a log-normal LN(0,σ²) distribution when σ² increases: blue stands for the empirical mean, while gold corresponds to the plug-in estimator exp(σ²/2) when σ² is estimated from the log-sample. (The sample is of size 10⁶.)

The question came on X validated and my first reaction was to doubt the implementation which outcome was so counter-intuitive. But then I thought about the representation of a log-normal variate as exp(σξ) when ξ is a standard Normal variate. When σ grows large enough, it is near impossible for σξ to be larger than σ². More precisely,

P(X>E[X])=P(σξ>σ²/2)=1-Φ(σ/2)

which can be arbitrarily small.

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