Truncation (statistics)

In statistics, truncation results in values that are limited above or below, similar to but distinct from the concept of statistical censoring.

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 and loss amounts that are actually above "u" are reported to the insurance company as being exactly "u" because "u" is the amount the insurance companies pay. The insurance company knows that the actual loss is greater than "u" but they don't know what is 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. Any loss amount that is greater than "d" will be reported to the insurance company is loss — "d" 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 a specific amount. They don't know how many losses occur or how much each loss is.

Probability distributions

Truncation can be applied to any probability distribution and will lead to a new distribution, not usually one within the same family. Thus, if a random variable "X" has "F"("x") as its distribution function, the new random varable "Y" defined as having the distribution of "X" trancated to the interval [a,b] has the distribution function:$F\_Y(y)=frac\{F(y)-F(a)\}\{F(b)-F(a)\}\; ,$for "y" in the interval [a,b] , and 0 or 1 otherwise.

Data analysis

The analysis of data where observations are treated as being from truncated versions of standard distributions can be undertaken using a maximum likelihood, where the likelihood would be derived from the distribution or density of the truncated distribution. This involves taking account of the factor $\{F(b)-F(a)\}$ in the modified density function which will depend on the parameters of the original distribution.

In practice, if the fraction truncated is very small the effect of truncation may be ignored when analysing data. For example, it is common to use a normal distribution to model data whose values can only be positive but for which the typical range of values is well away from zero: in such cases a truncated or censored version of the normal distribution may formally be preferable (although there would be other alternatives also), but there would be very little change in results from the more complicated analysis.

ee also

*Truncated distribution
*Truncated mean
*Censoring (statistics)