good terminology for the parameters of a lognormal distribution?

Question

Is there any good short terminology for the two parameters of a lognormal distribution?

I have been using mean-log for $\mu$ and volatility for $\sigma$, where the lognormal variable $X$ has $\ln(X)$ distributed normally with mean $\mu$ and standard deviation $\sigma$.

This terminology is not standard, but then neither Mathworks nor Mathworld nor NIST nor Wikipedia seems to have any standard phrases of similar length for these quite common parameters.

1. Mean-log has the advantages of being short, clear, and parallel to the well-established meaning of "mean square". It may not be parallel to the word "lognormal", but I still find it preferable to the alternative:

   • "log-mean" would suggest $\ln(E(X))$.

2. Volatility works well for one common use of the lognormal distribution, e.g. modeling relative stock prices in one year by a lognormal distribution with mean-log 8.5% and volatility 19.5%. But this does not work so well for lognormal models of StackExchange votes or storm sizes. On the other hand, I find this preferable to the convolutedness or awkwardness of the alternatives:

   • "modeling relative stock prices in one year by a lognormal distribution whose log has mean 8.5% and standard deviation 19.5%"
   • "standard deviation of the log of the variable of 19.5%"
   • "s.d.-log of 19.5%"
   • "shape of 19.5%" (would one ever say that a normal distribution had a shape of 19.5%?)
   • "sigma of 19.5%"

Does any one have any better short terminology to suggest?

Answer 1

While not necessarily anything like a real convention, when I have to name the parameters at all (beyond "mu" and "sigma") I tend to call them the log-mean and log-standard-deviation (and I have seen at least a few other people do something similar), or sometimes "mean on the log scale" and "standard deviation on the log scale".

I also sometimes refer to them respectively as the "scale parameter" and the "shape parameter" (though more strictly exp(μ) is a scale parameter rather than μ itself, most people seem to catch on to the intent).