# Lognormal with negative values

I have some logged increments from time series data and wanted to fit a lognormal distribution, but obviously some are negative. How can I do this?

• If you have negative values, your data aren't lognormal. It may be that some modification is feasible (e.g. a mixture of lognormal and something else, or a lognormal location-mixture of, say, normals, or a shifted lognormal, or ...), but the lognormal itself isn't possible. But if the original values are lognormal, logged values would be fitted by a normal, which handles negatives. Can you more explicitly show what you mean by 'logged increments'? I can see two ways to interpret it and only one (the less likely one) makes a lot of sense for this, I think. – Glen_b Mar 4 '14 at 22:06

## 1 Answer

I think you've confused what the lognormal distribution is. If Y follows a lognormal distribution, then log(Y) follows a normal distribution. You seem to be thinking, incorrectly, that if Z follows a normal distribution, then log(Z) follows a lognormal distribution, which is not the case. It's an easy mistake to make.

In time series analysis, you often assume that the variable you're tracking, call it Y, changes multiplicatively. That is, $Y_{i+1}=Y_i * e ^ \delta$. When taking the log of both sides you'll see that $\log Y_{i+1} = \log Y_i + \delta$. This is why we talk about log increments. Now we assume that $\delta$ is normally distributed, which would make $e^\delta$ follow a lognormal distribution.