Advanced Loss Distributions and Tail Behavior

Tail Classification

Loss distributions are classified by tail weight. Light-tailed distributions (exponential, normal) have moment generating functions that exist everywhere. Heavy-tailed distributions have MGFs that do not exist for all positive t. Subexponential distributions (Pareto, lognormal, Weibull with tau < 1) satisfy P(X_1 + X_2 > x) ~ 2*P(X_1 > x), meaning the sum is driven by the maximum. Regularly varying distributions have P(X > x) ~ L(x)*x^{-alpha} for a slowly varying function L, with the Pareto as the canonical example. The tail index alpha determines which moments exist.

Tail Estimation and Diagnostics

The mean excess plot e(d) = E[X-d|X>d] versus d is a key diagnostic. A linear, increasing mean excess plot suggests a generalized Pareto tail. The Hill estimator estimates the tail index from the k largest observations: alpha-hat = (1/k) * sum of ln(X_{(n-i+1)}/X_{(n-k)}) for i=1 to k. QQ plots against exponential or Pareto references assess tail fit. For Exam MAS-II, understanding tail behavior is crucial for risk measure calculation, reinsurance pricing, and capital modeling. Know how to identify, estimate, and model heavy tails.