Loss Models: From Data to Decisions, 4th Edition

A.4 Distributions for large losses

The general form of most of these distributions has probability starting or ending at an arbitrary location. The versions presented here all use zero for that point. The distribution can always be shifted to start or end elsewhere.

A.4.1 Extreme value distributions

A.4.1.1 Gumbel—θ, μ (μ can be negative)

A.4.1.2 Frechet—α, θ This is the inverse Weibull distribution of Section A.3.2.4.

A.4.1.3 Weibull—α, θ³

A.4.2 Generalized Pareto distributions

A.4.2.1 Generalized Pareto—γ, θ This is the Pareto distribution of Section A.2.3.1 with α replaced by 1/γ and θ replaced by αθ.

A.4.2.2 Exponential—θ This is the same as the exponential distribution of Section A.3.3.1 and is the limiting case of the above distribution as γ → 0.

A.4.2.3 Pareto—γ, θ This is the single-parameter Pareto distribution of Section A.5.1.4. From the above distribution, shift the probability to start at θ.

A.4.2.4 Beta—α, θ This is the beta distribution of Section A.6.1.2 with a = 1. ...