method of moments
/ METH-ud uv MOH-ments /
Suppose you have a bag of data and a candidate formula (a distribution) with one or two unknown dials to set. The method of moments is the simplest sensible way to set those dials: make the distribution's theoretical averages match the data's actual averages. If your data's average is 5, pick the dial so the distribution's mean is also 5. It is the statistical equivalent of tuning a recipe until it tastes like the dish you remember.
Moments are summary numbers of a distribution: the first moment is the mean, the second is related to the variance, and so on. The method works by writing the distribution's moments as formulas in the unknown parameters, then setting each theoretical moment equal to the corresponding sample moment and solving. For a distribution with one parameter you match one moment; with two parameters, two moments. For example, the exponential distribution has mean 1 / lambda. If the sample average claim wait is 4 days, you set 1 / lambda = 4 and get the estimate lambda = 0.25. With two-parameter families like the gamma you match both the sample mean and the sample variance and solve the pair of equations.
Actuaries reach for the method of moments because it is fast, requires no calculus or computer search, and almost always gives an answer — handy for a quick first fit of a loss-severity or claim-frequency distribution. Its weakness is that it usually throws away information and so produces estimates that wobble more than the gold-standard maximum likelihood estimates. It can even give nonsensical results, such as a negative variance estimate. In practice it is often used to generate good starting values that a maximum likelihood routine then refines.
Fitting a gamma to claim sizes, an actuary computes the sample mean ($3,000) and sample variance (18,000,000). Since the gamma's mean is alpha*theta and its variance is alpha*theta^2, solving the pair gives theta = 6,000 and alpha = 0.5 — a fit obtained with no computer search at all.
Match the mean and variance, solve, done — the appeal of moment matching.
Method-of-moments estimates are quick but statistically inefficient and can fall outside the valid parameter range; for serious fits they are usually a starting point, not the final word.