Method of Moments

Before a desk fits a return distribution by numerically maximising a likelihood — the subject of the next lesson — it usually wants a fast, closed-form starting guess: a rough shape and scale for a loss-severity model, an overdispersion parameter for a trade-count model, or initial values to hand to an iterative optimiser so it converges to the right root rather than wandering. The method of moments (MOM) is the oldest tool for exactly this job. It asks a deliberately simple question — for what parameter value do the distribution's theoretical moments match the moments actually observed in the sample? — and turns it into an algebraic recipe: write the low-order population moments as functions of the unknown parameters, invert those functions, and plug in the sample moments. No likelihood, no calculus of a log-density, no numerical search.

This note opens Topic 3 of the Statistics I unit by giving MOM a precise statement, working it through the Normal, Gamma, Beta, and Negative Binomial families, proving the consistency that makes MOM more than a curve-fitting trick, and flagging the pitfall — estimates that fall outside the parameter space — that MOM's simplicity buys at the price of robustness. The next lesson takes up maximum likelihood, which fixes MOM's efficiency shortfall at the cost of needing a numerical solve — for which, as the Gamma example below shows, the MOM estimate is the natural starting value.

Learning objectives. By the end of this note you should be able to: