Loss Distributions

• Non-negativity
• Right-skewed
• Heavy Tails
• Overdispersion
• Censoring & Truncation
• Discreteness & Continuity
• Heterogeneity
• Non-normal

Exponential distribution

• interarrival time
• memoryless: \(X\sim\mathcal{E}(\lambda)\quad\to\quad X-w\ |\ X>w\sim\mathcal{E}(\lambda)\)

Gamma distribution

• \(\mathcal{E}(\lambda)=\mathcal{G}(1,\lambda)\)
• sum of \(\alpha\) iid \(\mathcal{E}(\lambda)\) is \(\mathcal{G}(\alpha, \lambda)\)
• sum of \(n\) iid \(\mathcal{G}(\alpha,\lambda)\) is \(\mathcal{G}(n\alpha,\lambda)\)

Lognormal distribution

• \(X\sim\mathcal{L}(\mu,\sigma^2)\quad\to\quad\ln(X)\sim\mathcal{N}(\mu,\sigma^2)\)

Pareto distribution

• \(X\sim\mathcal{P}(\alpha,\lambda)\quad\to\quad X-w\ |\ X>w\sim\mathcal{P}(\alpha,\lambda+w)\)