Estimate the parameters for frequency and severity distributions by maximum likelihood.

MLE is the default estimator on this exam. Either the score equation solves in closed form (Poisson, exponential, lognormal) or you log-likelihood-and-Newton (gamma, Weibull, Pareto, negative binomial).

Building the likelihood. Each policy contributes one factor based on how it was observed.

• Complete observation at $x_i$: factor $f(x_i;\theta )$.
• Right-censored at policy limit $u_i$ (claim exceeded the cap, you only know $X_i>u_i$): factor $S(u_i;\theta )$.
• Left-truncated at deductible $d_i$ (claims below $d_i$ never reported): divide by $S(d_i;\theta )$.
• Interval-censored on $[a_i,b_i]$: factor $F(b_i)-F(a_i)$.

Multiply across all observations to get $L(\theta )$. Take logs to get $\ell (\theta )=\ln L(\theta )$. Differentiate, set to zero, solve.

$$L(\theta )=\prod _{i\in C}\frac{f(x_i;\theta )}{S(d_i;\theta )}\cdot \prod _{j\in R}\frac{S(u_j;\theta )}{S(d_j;\theta )}$$

KEY: Truncation divides by $S(d)$; censoring substitutes $S(u)$ for $f$. Mixing them up is the single most punished error in this LO.

Common mistakes

• Dividing exponential MLE by total observations including censored ones. Use only complete count in the denominator.
• Using divisor $n-1$ for lognormal ${\hat{\sigma }}^2$. MLE uses $n$; $n-1$ is the unbiased estimator, not the MLE.
• Forgetting to divide by $S(d)$ on left-truncated claims. Builds in deductible bias.

Bottom line

• Likelihood is the product of $f(x_i)$ over complete observations, $S(u_i)$ over right-censored ones, and $f(x_i)/S(d_i)$ over left-truncated ones.
• Closed-form severity: exponential $\hat{\theta }=\bar{x}$; lognormal $\hat{\mu }=\overline{\ln x}$ and ${\hat{\sigma }}^2=\overline{(\ln x-\hat{\mu }{)}^2}$ with divisor $n$, not $n-1$.
• Closed-form frequency: Poisson $\hat{\lambda }=\bar{n}$; binomial $\hat{q}=\bar{n}/m$.
• Right censoring: exponential $\hat{\theta }=(\sum x_i)/n_C$ divides by the complete count $n_C$, never the full $n$.

Exam shortcut

Exponential with censoring: add ALL claim amounts (complete and capped), divide by COUNT of complete claims. Pareto MLE with known $\theta$ and truncation at $d$: $\hat{\alpha }=n/\sum \ln [(x_i+\theta )/(d+\theta )]$. The anchor in the denominator is always $d+\theta$. Lognormal: transform first to $y_i=\ln x_i$, then take the population mean and population variance. Convert back only at the end via $E[X]=\exp (\hat{\mu }+{\hat{\sigma }}^2/2)$. Exponential $\ell (\hat{\theta })=-n\ln \bar{x}-n$. Memorize the trailing $-n$. Bernoulli-from-exponential: $\hat{\beta }=-(1/t_0)\ln ((n-M)/n)$.

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Learning objectives

• 4a

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