We say that a random variable X has a lognormal distribution if its logarithm, Y = log X, is normally distributed. The normal distribution often occurs when a random variable comes about by combining a bunch of small independent contributions, but those contributions combine additively; when the combination is multiplicative instead, lognormals occur. For example, lognormal distributions often occur in models of financial markets.
But of course X = exp Y, so the variable we care about is the exponential of a normal. Why isn't it called expnormal?
I don't know if you've seen this paper, but they make the case that it's actually the lognormal distribution that's ubiquitous in primary experimental data. In biology at least, I think it makes sense, since most features are generated by growth processes where errors tend to have multiplicative rather than additive effects.
ReplyDeleteMike,
ReplyDeleteI hadn't seen that, and lack the time to read it right now, but thanks for the pointer. The central point seems reasonable at first glance; when the variance is small the lognormal approaches the normal, so it's easy to get them confused.
I agree that it seems backward. You might reasonably think that lognormal means "log of a normal." But instead it means "distribution whose log is normal."
ReplyDeleteAlthough this is confusing, it is conventional in probability. A [foo][bar] distribution is one such that when you apply function [foo], you get distribution [bar]. For example, logit-normal is sometimes used for a distribution such that its logit is normal, i.e. the inverse logit of a normal distribution.
I'm guessing lognormal means that it looks like a normal distribution on log graph paper.
ReplyDeleteAnybody else remember log graph paper or am I dating myself.
It's the same pattern as log-convex. In general I feel like saying X is noun-adjective should mean that the noun of X is adjective, but I'm having trouble coming up with good real-world examples.
ReplyDeleteDoes that pattern sound right to you when the noun is not the name of a function like log? For example, if someone were to coin the word "derivative-increasing", would you interpret it as nonnegative or convex? I'd definitely choose convex, but maybe it's just me.