Statistical dead end

I get suspicious when I hear people ask about third and fourth moments (skewness and kurtosis). I’ve heard these terms far more often from people who don’t understand statistics than from people who do.

There are two common errors people often have in mind when they bring up skewness and kurtosis.

First, they implicitly believe that distributions can be boiled down to three or four numbers. Maybe they had an elementary statistics course in which everything boiled down to two moments — mean and variance — and they suspect that’s not enough, that advanced statistics extends elementary statistics by looking at third or fourth moments. “There are more things in heaven and earth, Horatio, than are dreamt of in your philosophy.” The path forward is not considering higher and higher moments.

This leads to a second and closely related problem. Interest in third and fourth moments sounds like hearkening back to the moment-matching approach to statistics. Moment matching was a simple idea for estimating distribution parameters:

  1. Set population means equal to sample means.
  2. Set population variances equal to sample variances.
  3. Solve the resulting equations for distribution parameters.

There’s more to moment matching that that, but that’s enough for this discussion. It’s a very natural approach, which is probably why it still persists. But it’s also a statistical dead end.

Moment matching is the most convenient approach to finding estimators in some cases. However, there is another approach to statistics that has largely replaced moment matching, and that’s maximum likelihood estimation: find the parameters that make the data most likely.

Both moment matching and maximum likelihood are intuitively appealing ideas. Sometimes they lead to the same conclusions but often they do not. They competed decades ago and maximum likelihood won. One reason is that maximum likelihood estimators have better theoretical properties. Another reason is that maximum likelihood estimation provides a unified approach that isn’t thwarted by difficulties in solving algebraic equations.

There are good reasons to be concerned about higher moments (including fractional moments) though these are primarily theoretical. For example, higher moments are useful in quantifying the error in the central limit theorem. But there are not a lot of elementary applications of higher moments in contemporary statistics.