Estimation, Evaluation, and Selection of Actuarial Models

6 CHAPTER 2. MODEL ESTIMATION
2.2 Estimation using data-dependent distributions
2.2.1 Introduction
When observations are collected from a probability distribution, the ideal situation is to have the
(essentially) exact 1 value of each observation. This case is referred to as “complete, individual
data.” This is the case in Data Sets B and D1. There are two reasons why exact data may not
be available. One is grouping, in which all that is recorded is the range of values in which the
observation belongs. This is the case for Data Set C and for Data Set A for those with five or more
accidents.
A second reason that exact values may not be available is the presence of censoring or truncation.
When data are censored from below, observations below a given value are known to be below that
value, but the exact value is unknown. When data are censored from above, observations above a
given value are known to be above that value, but the exact value is unknown. Note that censoring
effectively creates grouped data. When the data are grouped in the first place, censoring has no
effect. For example, the data in Data Set C may have been censored from above at 300,000, but
we cannot know for sure from the data set and that knowledge has no effect on how we treat the
data. On the other hand, were Data Set B to be censored at 1,000, we would have fifteen individual
observations and then five grouped observations in the interval from 1,000 to infinity.
In insurance settings, censoring from above is fairly common. For example, if a policy pays no
more than 100,000 for an accident, any time the loss is above 100,000 the actual amount will be
unknown, but we will know that it happened. In Data Set D2 we have random censoring. Consider
the fifth policy in the table. When the “other information” is not available, all that is known about
the time of death is that it will be after 1.8 years. All of the policies are censored at 5 years by the
nature of the policy itself. Also, note that Data Set A has been censored from above at 5. This is
more common language than to say that Data Set A has some individual data and some grouped
data.
When data are truncated from below, observations below a given value are not recorded. Truncation
from above implies that observations above a given value are not recorded. In insurance
settings, truncation from below is fairly common. If an automobile physical damage policy has a
per claim deductible of 250, any losses below 250 will not come to the attention of the insurance
company and so will not appear in any data sets. Data Set D2 has observations 31—40 truncated
from below at varying values. The other data sets may have truncation forced on them. For example,
if Data Set B were to be truncated from below at 250, the first seven observations would
disappear and the remaining thirteen would be unchanged.
2.2.2 The empirical distribution for complete, individual data
AsnotedinDefinition 2.3, the empirical distribution assigns probability 1/n to each data point.
That works well when the value of each data point is recorded. An alternative definition is
Definition 2.5 The empirical distribution function is
number of observations ≤ x
F n (x) =
n
1 Some measurements are never exact. Ages may be rounded to the nearest whole number, monetary amounts
to the nearest dollar, car mileage to the nearest tenth of a mile, and so on. This Note is not concerned with such
rounding errors. Rounded values will be treated as if they are exact.