12.1 Point estimation

It is not unusual for data to be incomplete due to censoring or truncation. The formal definitions are as follows.

Definition 12.1 An observation is truncated from below (also called left truncated) at d if when it is at or below d it is not recorded, but when it is above d it is recorded at its observed value.

An observation is truncated from above (also called right truncated) at u if when it is at or above u it is not recorded, but when it is below u it is recorded at its observed value.

An observation is censored from below (also called left censored) at d if when it is at or below d it is recorded as being equal to d, but when it is above d it is recorded at its observed value.

An observation is censored from above (also called right censored) at u if when it is at or above u it is recorded as being equal to u, but when it is below u it is recorded at its observed value.

In insurance claim data, the most common occurrences are left truncation and right censoring. Left truncation occurs when an ordinary deductible of d is applied. When a policyholder has a loss below d, he or she realizes no benefits will be paid and so does not inform the insurer. When the loss is above d, the amount of the loss is assumed to be reported.1 A policy limit is an example of right censoring. When the amount of the loss equals or exceeds u, benefits beyond that value are not paid, and so the exact value is not recorded. However, it is known that a loss of at least u has occurred. ...

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