Truncation (statistics)

In statistics, truncation results in values that are limited above or below, resulting in a truncated sample. A random variable is said to be truncated from below if, for some threshold value , the exact value of is known for all cases , but unknown for all cases . Similarly, truncation from above means the exact value of is known in cases where , but unknown when . Truncation is similar to but distinct from the concept of statistical censoring. A truncated sample can be thought of as being equivalent to an underlying sample with all values outside the bounds entirely omitted, with not even a count of those omitted being kept. With statistical censoring, a note would be recorded documenting which bound (upper or lower) had been exceeded and the value of that bound. With truncated sampling, no note is recorded. Usually the values that insurance adjusters receive are either left-truncated, right-censored, or both. For example, if policyholders are subject to a policy limit u, then any loss amounts that are actually above u are reported to the insurance company as being exactly u because u is the amount the insurance company pays. The insurer knows that the actual loss is greater than u but they don't know what it is. On the other hand, left truncation occurs when policyholders are subject to a deductible. If policyholders are subject to a deductible d, any loss amount that is less than d will not even be reported to the insurance company. If there is a claim on a policy limit of u and a deductible of d, any loss amount that is greater than u will be reported to the insurance company as a loss of because that is the amount the insurance company has to pay. Therefore, insurance loss data is left-truncated because the insurance company doesn't know if there are values below the deductible d because policyholders won't make a claim. The insurance loss is also right-censored if the loss is greater than u because u is the most the insurance company will pay. Thus, it only knows that your claim is greater than u, not the exact claim amount.