Truncated and Censored Data in Loss Modeling

Types of Data Modification

Insurance data is rarely complete. Left truncation occurs when losses below a deductible d are unobserved (you only see losses exceeding d). Right censoring occurs at a policy limit u (you know a loss exceeds u but not the actual amount). These modifications affect likelihood construction. For a left-truncated observation x > d, the likelihood contribution is f(x)/S(d). For a right-censored observation at u, the contribution is S(u). Combined truncation and censoring means the contribution for an observed loss x is f(x)/(1-F(d)) and for a censored loss is S(u)/(1-F(d)).

Impact on Estimation

Ignoring truncation or censoring biases parameter estimates. With truncation, the mean of observed data overstates the population mean for the unmodified distribution. MLE handles these data complications naturally through proper likelihood construction. Exam STAM frequently presents data sets with deductibles and limits, asking you to write the correct likelihood, find MLE estimates, and understand how policy modifications create truncation and censoring in loss data.