The geometric mean is the appropriate measure of central tendency for log-normally distributed variables. However, the arithmetic mean still has some use in relation to log-normal variables - in inferring totals from survey data, for instance. It is my understanding that the arithmetic mean of the sample of a log-normal distribtion is the same as the arithmetic mean of the population, with normal errors providing the sample is large enough for CLT to be invoked.
If you operate a simple missing data imputation method of replacing with a mean, you would inflate the geometric mean if you used the arithmetic mean (and give high leverage to imputed data), and deflate the arithmetic mean (and therefore the total) if you used the geometric mean.
My question is what is most appropriate to use, the geometric mean or the arithmetic mean? Or is such a simple imputation method completely inappropriate for log-normal data?