Method of Moments Estimation for Insurance Losses

The Method

The method of moments (MOM) estimates parameters by equating sample moments to theoretical moments. For a distribution with k parameters, set the first k sample moments equal to the first k theoretical moments and solve. The j-th sample moment is m_j = (1/n) * sum(xi^j) and the j-th theoretical moment is mu_j'(theta) = E[X^j].

For the exponential: E[X] = theta, so theta_hat = x_bar (same as MLE in this case). For the lognormal: E[X] = exp(mu + sigma^2/2) and E[X^2] = exp(2*mu + 2*sigma^2). Setting these equal to the sample moments and solving gives mu_hat and sigma_hat.

Advantages and Disadvantages

MOM is simple to compute and always produces a closed-form (or easily solved) estimator. However, MOM estimators are generally less efficient than MLEs (higher variance). MOM can produce estimates outside the parameter space (e.g., negative variance estimates). MOM does not naturally handle censored or truncated data, though adjustments using empirical limited moments are possible.

Percentile Matching

An alternative to matching moments is percentile matching: set theoretical percentiles equal to sample percentiles and solve for parameters. For example, for a two-parameter distribution, equate the theoretical median and 90th percentile to the empirical median and 90th percentile. This approach is useful when moments do not exist (heavy-tailed distributions) or when tail fit is more important than overall fit. For Exam STAM, be comfortable with both moment matching and percentile matching. Problems typically give sample statistics and ask you to estimate parameters for a specified distribution using the method indicated.