Estimation of Truncated Data Samples in Operational Risk Modeling

This article addresses challenges of estimating operational risk regulatory capital when a loss sample is truncated from below at a data collection threshold. Recent operational risk literature reports that the attempts to estimate loss distributions by the maximum likelihood method are not always successful under the truncation approach that accounts for the existence of censored losses—the likelihood surface is sometimes ascending with no global solution. The literature offers an alternative called the shifting approach, which estimates the loss distribution without taking into account censored losses. We present a necessary and sufficient condition for the existence of the global solution to the likelihood maximization problem under the truncation approach when the true loss distribution is lognormal, and derive a practically explicit expression for the global solution. We show by a simulation study that, as the sample size increases, the capital bias by the truncation approach declines while the bias by the shifting approach does not.