Likelihood for Truncated Loss Data¶

Definition¶

Under left truncation at $d$, only losses satisfying $X>d$ enter the dataset. For independent exact observations drawn from that conditional population, the likelihood contribution is $f_\theta(x_i)/\bar F_\theta(d)$ for $x_i>d$$d$, provided $\bar F_\theta(d)>0$$0$.

Interpretation¶

Small losses are absent altogether. The likelihood must renormalize the observed population; this differs from retaining small losses as censored observations.

The figure uses the fitted rate $2/70$ from the Examples case. The dashed ground-up density $f_\theta$ puts almost all of its mass below $d=100$$100$, where nothing is recorded. Dividing by $\bar F_\theta(100)=\mathrm{e}^{-20/7}\approx0.057$ lifts the part above 100 into a proper density on $x>100$$100$, and the observations 120 and 150 contribute its heights $f_\theta(x_i)/\bar F_\theta(d)$ to the likelihood. For the exponential law this conditional density is the original one shifted to start at $d$, which is why only the excesses 20 and 50 enter the estimate.

An ordinary deductible produces a dataset of paid claims only when the loss exceeds $d$. If recorded amounts are excesses, add $d$ back before evaluating the ground-up density, and divide by the exceedance probability to describe the selected population.

Key Results¶

• $\displaystyle \ell(\theta)=\sum_i\log f_\theta(x_i)-n\log\bar F_\theta(d)$ for a fixed observed sample size.
• For observation restricted to $(a,b]$, normalize by $F_\theta(b)-F_\theta(a)$$F_{\theta }(a)$.
• If a deductible dataset reports excesses $y_i=X_i-d$$d$, use $f_\theta(y_i+d)/\bar F_\theta(d)$$d)/{\bar{F}}_{\theta }(d)$.
• Information about the total number of excluded losses would require an additional sampling model.

Examples¶

For exponential losses observed only above 100, observations 120 and 150 have excesses 20 and 50. The conditional maximum likelihood estimate (MLE) is $2/(20+50)=2/70$$2/70$, rather than $2/270$.

Prerequisites¶

• Maximum Likelihood Estimation
• Ordinary Deductible

Related Concepts¶

• Likelihood for Censored Loss Data
• Likelihood for Grouped Loss Data

Sources¶

• Risk Insurance · Lecture 5

Aliases: left truncation, deductible-conditioned sampling, truncated likelihood, truncated data likelihood, left-truncated data Keywords: f_θ(x_i)/F̄_θ(d), deductible data, excesses over the deductible, conditional MLE Symbols: sum, theta

Last updated: 17 October 2026 21:53