Estimation, Evaluation, and Selection of Actuarial Models

12 CHAPTER 2. MODEL ESTIMATION
Exercise 2 Construct the ogive and histogram for the following data set.
Payment range Number of payments
0—25 6
25—50 24
50—75 30
75—100 31
100—150 57
150—250 80
250—500 85
500—1000 54
1000—2000 15
2000—4000 10
over 4000 0
2.2.4 Empirical distributions for censored and truncated data
It is not unusual for data to be incomplete due to censoring or truncation. The formal definitions
are as follows.
Definition 2.15 An observation is truncated from below (also called left truncated) atd if,
when it is 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) atu if, when it is 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) atd if, when it is 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) atu if, when it is above u,
it is recorded as being equal to u, butwhenitisbelowu, itisrecordedatitsobservedvalue.
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, heorshe
knows no benefits will be paid and so does not inform the insurer. When the loss is above d, the
amount of the loss will be reported. A policy limit is an example of right censoring. When the
amount of the loss 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.
When constructing a mortality table, it is impractical to follow people from birth to death. It
is more common to follow a group of people of varying ages for a few years. When a person joins
a study, he or she is alive at that time. This person’s age at death must be at least as great as the
age at entry to the study, and thus has been left truncated. If the person is alive when the study
ends, right censoring has occurred. The person’s age at death is not known, but it is known that it
is at least as large as the age when the study ended. Right censoring also affects those who leave
the study prior to its end due to surrender.
Because left truncation and right censoring are the most common occurrences in actuarial work,
they are the only cases that will be covered in this Note. To save words, “truncated” will always
mean truncated from below and “censored” will always mean censored from above.
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