Likelihood for Censored Loss Data¶

Definition¶

Under right censoring at a known cap $c_i$, an exact continuous loss $x_i<c_i$ is observed if it is below the cap; otherwise one observes only $X_i\ge c_i$$c_i$. With indicator $\delta_i=1$ for an exact observation, its likelihood contribution is $f_\theta(x_i)^{\delta_i}\bar F_\theta(c_i)^{1-\delta_i}$ under continuous losses and non-informative censoring (the censoring caps are independent of the losses and carry no information about $\theta$).

Interpretation¶

The policy exists in the sample even when its full loss is unknown. A capped observation gives a tail event, not a point density at the cap.

The curve is the exponential density at the Example's estimate $\hat\lambda=2/150$$2/150$. The exact losses 20 and 30 enter the likelihood through the heights $f_\theta(20)$ and $f_\theta(30)$; the observation capped at $c_i=100$ enters through the full tail probability $\bar F_\theta(100)=\mathrm{e}^{-100\lambda}$${\mathrm{e}}^{-100\lambda }$, which at $\hat\lambda$ equals $\mathrm{e}^{-4/3}\approx0.26$$0.26$. The shading shows only losses from 100 to 205: its probability is $\mathrm{e}^{-4/3}-\mathrm{e}^{-41/15}\approx0.19859$$0.19859$, while the omitted tail above 205 has probability $\mathrm{e}^{-41/15}\approx0.06500$$0.06500$. The product $\lambda^2\mathrm{e}^{-50\lambda}\cdot \mathrm{e}^{-100\lambda}$ has logarithm $2\log\lambda-150\lambda$$150\lambda$, the log-likelihood of the Example.

Maximum likelihood estimation maximizes the probability or density of the data actually recorded. A capped observation records a tail event, so its contribution is the survivor function at the cap rather than the density of an exact loss there.

Key Results¶

• Independent contributions multiply; exact observations contribute log densities and censored observations log survivor probabilities.
• For a common exponential rate, $\displaystyle \hat\lambda=r/[\sum_{\mathrm{exact}}x_i+\sum_{\mathrm{censored}}c_i]$$\sum _{\mathrm{censored}}{c}_i]$, if $r>0$ is the number of exact losses.
• If no exact losses occur, an interior maximum need not exist.
• An atom at the censoring boundary requires careful use of strict or weak tail events.

Examples¶

For exact losses 20 and 30 plus one observation capped at 100, exponential log-likelihood is $2\log\lambda-150\lambda$$150\lambda$. Thus $\hat\lambda=2/150$$2/150$.

Prerequisites¶

• Maximum Likelihood Estimation
• Survivor Function

Related Concepts¶

• Policy Limit
• Likelihood for Truncated Loss Data
• Likelihood for Grouped Loss Data

Sources¶

• Risk Insurance · Lecture 5

Aliases: right censoring, policy-limit censoring, censored likelihood, censored data likelihood Keywords: censoring indicator, non-informative censoring, survivor function at the cap, capped losses, exponential MLE with censoring Symbols: delta, lambda, sum, theta

Last updated: 27 October 2026 11:52