Advanced Loss Modeling

A loss data set with substantial right tail is given and you must fit a heavy-tailed distribution and compute tail quantities.

Plot the data on a log-log scale or use a mean excess plot to assess tail thickness. Fit candidate distributions (Pareto, Weibull, lognormal, generalized beta of the second kind) by MLE. Compare fits with AIC/BIC and Anderson-Darling (tail-weighted). Use the fitted model to compute VaR, TVaR, or layered loss costs.

Mean excess function e(x) = E[X - x | X > x]; for Pareto, e(x) grows linearly in x.

Loss data is right-censored by a policy limit and left-truncated by a deductible, and you must compute MLE for the severity distribution.

Write the likelihood with each observation's correct contribution: f(x)/S(d) for ground-up losses observed in full, S(u)/S(d) for losses censored at the limit u with deductible d. Take logs and maximize numerically. Recognize that ignoring truncation biases parameter estimates upward in the tail.

L = Π_i [f(x_i)/S(d)]^{δ_i} [S(u)/S(d)]^{1-δ_i}; δ_i = 1 if fully observed, 0 if censored.

You must model the very high-loss tail using EVT methods such as the generalized Pareto distribution (GPD) on excesses over a threshold.

Pick a threshold u above which the data look heavy-tailed (use a mean excess plot). Fit the GPD to the excesses X - u given X > u. Use the fitted shape parameter ξ to characterize the tail: ξ > 0 → heavy (Pareto-like), ξ = 0 → exponential-like, ξ < 0 → bounded. Compute VaR and TVaR using the fitted GPD.

GPD excess distribution: F(y) = 1 - (1 + ξ y / σ)^{-1/ξ}; shape ξ characterizes tail.