Maximum Likelihood Estimation for Loss Distributions

MLE with Complete Data

For a random sample x1, ..., xn from a distribution f(x|theta), the log-likelihood is l(theta) = sum of ln(f(xi|theta)). The MLE maximizes this. For the exponential: l(theta) = -n*ln(theta) - sum(xi)/theta, giving theta_hat = x_bar. For the Pareto(alpha, theta) with theta known: l(alpha) = n*ln(alpha) + n*alpha*ln(theta) - (alpha+1)*sum(ln(xi + theta)), giving alpha_hat = n / sum(ln((xi + theta)/theta)).

For multi-parameter distributions, solve the system of score equations simultaneously (often numerically).

Censored and Truncated Data

Right-censored data: for an observation censored at c, the contribution to the likelihood is S(c|theta) = 1 - F(c|theta) instead of f(x|theta). Left truncation at d: observations are only recorded if X > d. The contribution of an observed value x is f(x|theta) / S(d|theta). The likelihood for data with both truncation and censoring is: L = product over uncensored of [f(xi) / S(di)] * product over censored of [S(ci) / S(di)], where di is the truncation point for observation i.

Exam STAM Applications

Insurance data frequently involves deductibles (left truncation) and policy limits (right censoring). The deductible means we only observe losses exceeding d, creating truncation at d. The policy limit means losses exceeding the limit are recorded at the limit, creating censoring. For Exam STAM, construct the likelihood correctly for each data type, differentiate, and solve. Common exam problems involve exponential, Pareto, or Weibull distributions with incomplete data. Fisher information and variance of the MLE are also tested: Var(theta_hat) is approximately 1/(n * I(theta)), where I is the Fisher information per observation.