Severity Distributions: Exponential, Pareto, Lognormal, Weibull for STAM

Exponential and Pareto

The exponential distribution with parameter theta has PDF f(x) = (1/theta)*e^(-x/theta), mean theta, and variance theta^2. It is light-tailed and memoryless. The (single-parameter) Pareto with parameters alpha and theta has PDF f(x) = alpha*theta^alpha/(x+theta)^(alpha+1), mean theta/(alpha-1) for alpha > 1. The Pareto is heavy-tailed: its moments exist only for orders less than alpha. The Pareto is widely used for large liability claims and reinsurance pricing.

The two-parameter Pareto (Pareto Type II or Lomax) is the standard in actuarial loss modeling. Its survival function S(x) = (theta/(x+theta))^alpha makes tail probability calculations straightforward.

Lognormal and Weibull

If ln(X) ~ Normal(mu, sigma^2), then X is lognormal with E[X] = exp(mu + sigma^2/2). The lognormal is moderately heavy-tailed and is commonly used for property and casualty losses. The Weibull has survival function S(x) = exp(-(x/theta)^tau). When tau > 1, the hazard rate increases (aging); when tau < 1, it decreases. The Weibull includes the exponential (tau = 1) as a special case and is used for warranty claims and mechanical failure times.

Choosing a Severity Distribution

Selection depends on the tail behavior, the availability of data, and the intended use. Compare distributions using: the mean excess plot (increasing suggests heavy tails), the log-log plot of the empirical survival function (linear suggests Pareto), and formal goodness-of-fit tests. For Exam STAM, know the PDFs, CDFs, means, variances, and limited expected values E[min(X, d)] for each distribution. Limited expected values are essential for computing expected payments under deductibles and policy limits.