Adjust calculations for the effect of missing data values, including censoring and truncation.

A claims analyst pulls a loss file and notices three patterns: some claims are still open at the valuation date, some never enter the database because they fall under a deductible, and a handful of severity fields are blank. Each pattern distorts naive averages in a different way, and MAS-I rewards candidates who can name the distortion and correct for it.

Censoring versus truncation. Censoring means you know the observation exists but only see a bound. Truncation means the observation never appears in your dataset at all. A policy with a $500 deductible and $100,000 limit produces left-truncated losses (below $500 you see nothing) that are also right-censored at $100,000 (you see "paid the limit" but not the true severity).

KEY: Censoring contributes partial information to the likelihood. Truncation removes the observation entirely and forces you to condition on the survival region.

• Right-censoring at $u$: you observe $Y=\min (X,u)$ and $\delta =1\{X\le u\}$. Uncensored points contribute $f(y)$; censored points contribute $S(u)=P(X>u)$.
• Left-censoring at $\ell$: you observe $Y=\max (X,\ell )$. Points at the bound contribute $F(\ell )=P(X\le \ell )$.

Common mistakes

• Treating a policy limit as truncation. A claim paid at the $2,000,000 limit is right-censored, not truncated. It still appears in the data and contributes $S(u)$ to the likelihood.
• Forgetting the $S(d)$ denominator under left-truncation. Without dividing by $S(d)$, the likelihood treats the deductible as nonexistent and biases $\hat{\lambda }$ toward zero (mean too high).
• Counting truncated subjects in the Kaplan-Meier risk set before entry. A subject who entered at $d_i=12$ cannot be at risk at event time 12 under a strict-inequality convention. Entering them early inflates $\hat{S}$.

Bottom line

• Censoring keeps the observation but bounds its value; truncation removes it from the sample entirely.
• Right-censoring observes $\min (X,u)$, left-censoring observes $\max (X,\ell )$: these contribute $S(u)$ or $F(\ell )$ while uncensored points contribute $f(y)$.
• Left-truncation at $d$ excludes $X<d$; condition on $X>d$ by dividing the density by $S(d)$.
• Right-truncation at $u$ excludes $X>u$; divide by $F(u)$.

Exam shortcut

When a problem says "deductible," write left-truncation and add $/S(d)$ to every observation's contribution before doing anything else. When a problem says "limit" or "maximum benefit," write right-censoring and replace $f(y)$ with $S(u)$ for those points.

The full lesson (about 1,723 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• B9

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