Estimation, Evaluation, and Selection of Actuarial Models

86 CHAPTER 5. MODELS WITH COVARIATES
While there is nothing wrong with the first approach, it is not very interesting. It just asks
us to repeat the modeling approach of this Note over and over as we move from one combination
to another. The second approach is a single parametric model that can also be analyzed with
techniques already discussed, but is clearly more parsimonious. The third model’s hybrid nature
implies that additional effort will be needed to implement it.
The third model would be a good choice when there is no obvious distributional model for
a given individual. In the case of automobile drivers, the Poisson distribution is a reasonable
choice and so the second model may be the best approach. If the variable is time to death, a
data-dependent model such as a life table may be appropriate.
The advantage of the second and third approaches over the first one is that for some of the
combinations there may be very few observations. In this case, the parsimony affordedbythe
second and third models may allow the limited information to still be useful. For example, suppose
our task was to estimate the 80 entries in a life table running from age 20 through age 99 for four
gender/smoker combinations. Using model 1 above there are 320 items to estimate. Using model
3 there would be 83 items to estimate. 1
5.2 Proportional hazards models
A particular model that is relatively easy to work with is the Cox proportional hazards model.
Definition 5.2 Given a baseline hazard rate function h 0 (t) and values z 1 ,...,z p associated with
a particular individual, the Cox proportional hazards model for that person is given by the
hazard rate function
h(x|z) =h 0 (x)c(β 1 z 1 + ···+ β p z p )=h 0 (x)c(β T z)
where c(y) is any function that takes on only positive values, z =(z 1 ,...,z p ) T is a column vector
of the z-values (called covariates) andβ =(β 1 ,...,β p ) T is a column vector of coefficients.
In this Note, the only function that will be used is c(y) =e y . One advantage of this function is
that it must be positive. The name for this model is fitting because if the ratio of the hazard rate
functions for two individuals is taken, the ratio will be constant. That is, one person’s hazard rate
function is proportional to any other person’s hazard rate function. Our goal is to use the methods
of this Note to estimate the baseline hazard rate function h 0 (t) and the vector of coefficients β.
Example 5.3 Suppose the size of a homeowners fire insurance claim as a percentage of the home’s
value depends upon the age of the house and the type of construction (wood or brick). Develop a
Cox proportional hazards model for this situation. Also, indicate the difference between wood and
brick houses of the same age.
Let z 1 = age (a non-negative whole number) and z 2 =1if the construction is wood and z 2 =0
if the construction is brick. Then the hazard rate function for a given house is
h(x|z 1 ,z 2 )=h 0 (x)e β 1 z 1+β 2 z 2
.
1 There would be 80 items needed to estimate the survival function for one of the four combinations. The other
three combinations each add one more item–the power to which the survival function is to be raised.