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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).

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