Method of Moments Estimation for Insurance Losses

The Method of Moments

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

The method is straightforward: write down the theoretical moment expressions, substitute the sample moments, and solve for the parameters. For one parameter, this reduces to solving a single equation. For two parameters, solve a system of two equations in two unknowns.

MOM estimators are always consistent (they converge to the true parameter values as the sample size grows) under mild regularity conditions. This property follows from the law of large numbers applied to the sample moments.

MOM for Common Distributions

For the exponential with mean theta: E[X] = theta, so theta_hat = x_bar = m_1. The MOM estimator equals the MLE in this case, which is a special occurrence rather than a general rule.

For the Pareto(alpha, theta) with both parameters unknown: E[X] = theta/(alpha-1) and E[X^2] = 2*theta^2/((alpha-1)*(alpha-2)). Setting m_1 = theta/(alpha-1) and m_2 = 2*theta^2/((alpha-1)*(alpha-2)), divide the second equation by the first squared: m_2/m_1^2 = 2*(alpha-1)/(alpha-2). Solving for alpha gives alpha_hat = (2*m_2 - m_1^2)/(m_2 - m_1^2), and then theta_hat = m_1*(alpha_hat - 1).

For the lognormal(mu, sigma^2): E[X] = exp(mu + sigma^2/2) and E[X^2] = exp(2*mu + 2*sigma^2). Taking logarithms: ln(m_1) = mu + sigma^2/2 and ln(m_2) = 2*mu + 2*sigma^2. Solving: sigma^2_hat = ln(m_2) - 2*ln(m_1) = ln(m_2/m_1^2) and mu_hat = ln(m_1) - sigma^2_hat/2.

Advantages and Disadvantages

The main advantage of MOM is simplicity. The estimators are easy to compute and always produce a closed-form (or easily solved) solution. No numerical optimization is required, which is especially valuable when the likelihood function is difficult to maximize. MOM also provides reasonable starting values for numerical MLE algorithms.

However, MOM estimators are generally less efficient than MLEs, meaning they have larger variance for the same sample size. They do not use all the information in the data; only the first k moments are utilized, ignoring the rest of the distributional structure. MOM can also produce estimates outside the parameter space. For example, the formula for alpha_hat in the Pareto can yield a value less than 1, which would imply the mean does not exist, even though the sample clearly has a finite mean.

MOM does not naturally handle censored or truncated data, which is a significant limitation for insurance applications where deductibles and policy limits are common. Adjustments using empirical limited moments are possible but add complexity.

Percentile Matching

An alternative to matching moments is percentile matching: set theoretical percentiles equal to sample percentiles and solve for parameters. For a two-parameter distribution, equate two theoretical percentiles (e.g., the 50th and 90th) to the corresponding empirical percentiles and solve the system of equations.

Example: For the Pareto(alpha, theta), the p-th percentile is pi_p = theta*((1-p)^(-1/alpha) - 1). If the sample median is 200 and the 90th percentile is 800, set theta*(0.5^(-1/alpha) - 1) = 200 and theta*(0.1^(-1/alpha) - 1) = 800. Dividing the second by the first eliminates theta: (0.1^(-1/alpha) - 1)/(0.5^(-1/alpha) - 1) = 4. Solve this equation numerically for alpha, then back-substitute to find theta.

Percentile matching is useful when moments do not exist (for very heavy-tailed distributions where E[X^k] is infinite) or when tail fit is more important than overall fit. Matching high percentiles (90th, 95th, 99th) emphasizes the tail of the distribution, which is often the region of greatest interest in insurance applications such as reinsurance pricing or extreme value analysis.

Matching Limited Moments

For data subject to policy limits, limited moments E[min(X, d)] can be matched instead of raw moments. The empirical limited moment at limit d is (1/n)*sum(min(x_i, d)). Equating this to the theoretical limited expected value E[X ^ d] and solving provides parameter estimates that account for the censoring inherent in limited data.

For the exponential: E[X ^ d] = theta*(1 - e^(-d/theta)). If the empirical limited moment at d = 500 equals 350, solve theta*(1 - e^(-500/theta)) = 350 for theta. This equation requires numerical solution but correctly handles the fact that observations above 500 are capped.

Limited moment matching is a practical technique for insurance data where policy limits restrict the observable range. It provides a middle ground between the simplicity of standard MOM and the rigor of MLE with censored data.

Comparing MOM and MLE

For Exam STAM, understanding the differences between MOM and MLE is important. MLE is asymptotically efficient (smallest variance) and naturally handles censored and truncated data. MOM is simpler and does not require specifying the full likelihood. When both methods apply to complete data, MLE is generally preferred for its statistical properties, but MOM is a valid alternative that is easier to compute.

In some cases, the two methods give identical results. For the exponential, both yield theta_hat = x_bar. For the Poisson, both yield lambda_hat = x_bar. These coincidences occur because the sufficient statistics for these distributions are the sample moments.

In other cases, the results can differ substantially, especially for small samples or heavy-tailed distributions. The Pareto is a common example: MOM and MLE can give meaningfully different estimates of alpha when the sample size is moderate and the tail is heavy.

Study Tips and Common Pitfalls

The most common mistake is algebraic errors when solving the moment equations, especially for two-parameter distributions where the algebra can be involved. Write out the theoretical moments carefully and check your solutions by substituting back. Another frequent error is using the wrong formula for higher moments. E[X^2] is not (E[X])^2; the relationship is E[X^2] = Var(X) + (E[X])^2.

For Exam STAM, be comfortable with MOM for exponential, Pareto, lognormal, and gamma distributions. Practice both moment matching and percentile matching problems. At least 5 practice problems covering different distributions and estimation methods will build the fluency needed for exam day.