# 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. Mar 4 '14 at 22:06

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.