# Bias in Estimators of Lognormal

I am modelling a process distributed as a 2 parameter lognormal distribution; determining the parameters by maximum likelihood.

I have simulated the bias in the estimators (logmean and logsd) as well as the skewness. The logmean seems unbiased and the logsd and skewness seem asymptotically biased; underestimating these parameters.

Is there any closed form correction of the bias of these estimators?

Since the logmean and the logsd are just the mean and sd in the normal, they share the biases of those estimators in the normal case.

This, then, is well documented ...

1) the sample mean of a normal is unbiased for the population mean, $\mu$.

2) the sample s.d. is biased for $\sigma$. You say you're using ML, so is your formula for the sd the $n$-denominator sd of the logs (the actual ML estimator) or did you use the $n-1$-denominator version? (both are biased)

The corrections are not closed form unless you regard ratios of gamma functions as 'closed form'.

see here:

Why is sample standard deviation a biased estimator of $\sigma$?

I have used the rule of thumb there once or twice but mostly I stick to the $n-1$ denominator - though when working with lognormals or near-lognormals I do sometimes use the MLE ($n$ denominator). The times when I've used the rule of thumb is when someone wants a (nearly-)unbiased estimate of sigma for a normal. It doesn't happen often (what good is one of those?).

As for the estimate of skewness, it's a function of $\sigma$. You could use Taylor expansion to derive an approximate correction, but I'd first ask in respect of this whole exercise - why do you want an unbiased estimate in the first place?. Of course, ceteris paribus, you'd prefer unbiasedness to not, but the ceteris are decidedly not paribus; you give up something to get unbiasedness, and it's not immediately clear why that's worth giving up to get it.

there is a closed form, unbiassed estimator for central moments of distributions

• Welcome to the site, @efreet. CV aims to build up a fairly self-contained & permanent body of actionable information about statistics. As such, would you mind providing a brief summary of the info there & why you think it will answer the question so readers can decide whether it's what they want to pursue & in case of linkrot? – gung - Reinstate Monica Mar 22 '13 at 3:28