# Variance of the method of moments estimator for $\mu$ of log-normal distribution

Suppose the data has originated from a log-normal distribution with parameters $$\mu$$ and $$\sigma$$ (i.e. the mean and standard deviation of the underlying normal distribution). I have only the sample mean and sample variance of the data (but not of the logged-data).

Method of moments can be used to produce an estimator for $$\mu$$ (see e.g. Wikipedia) but is there a proper way of estimating the variance of this estimator (or simply producing a confidence interval for $$\mu$$)? The fact that the estimator is not very well behaved has been noted (see e.g. comment to a previous CV question or subsection 3.2.2 of a Masters Thesis).