Maximum Likelihood Estimation for Actuaries

The Likelihood Function

Given observed data x1, ..., xn from a distribution with parameter theta, the likelihood function is L(theta) = product of f(xi | theta). The maximum likelihood estimator (MLE) is the value of theta that maximizes L(theta), or equivalently, the log-likelihood l(theta) = sum of ln(f(xi | theta)). Taking logarithms converts products to sums, simplifying calculations.

To find the MLE: set the score function dl/d_theta = 0 and solve. Verify the solution is a maximum using the second derivative (should be negative) or by inspecting the likelihood shape.

Properties of MLEs

Under regularity conditions, MLEs have desirable large-sample properties: consistency (converges to the true parameter as n grows), asymptotic normality (theta_hat is approximately Normal(theta, 1/I(theta)) for large n, where I(theta) is the Fisher information), and asymptotic efficiency (achieves the Cramer-Rao lower bound). The invariance property states that the MLE of g(theta) is g(theta_hat) for any function g.

The Fisher information I(theta) = -E[d^2l/d_theta^2] = Var(dl/d_theta). The standard error of theta_hat is approximately 1/sqrt(n*I(theta)).

Applications in Actuarial Modeling

MLE is the primary estimation method for GLMs, logistic regression, and loss distribution fitting. For Exam SRM, you should be able to derive the MLE for common distributions (normal, Poisson, exponential, binomial), construct confidence intervals using the asymptotic normal approximation, and perform likelihood ratio tests. With censored or truncated data (common in insurance), the likelihood is modified to account for the incomplete observation, but the maximization principle remains the same.