Estimation, Evaluation, and Selection of Actuarial Models

5.2. PROPORTIONAL HAZARDS MODELS 87
One consequence of this model is that regardless of the age, the effect of switching from brick to
wood is the same. For two houses of age z 1 we have
The effect on the survival function is
·
S wood (x) = exp −
h wood (x) =h 0 (x)e β 1 z 1+β 2 = h brick (x)e β 2.
Z x
= [S brick (x)] exp(β 2 ) .
0
¸ ·
h wood (y)dy =exp −
Z x
0
¸
h brick (y)e β 2dy
¤
The baseline hazard rate function can be estimated using either a parametric model or a datadependent
model. The remainder of the model is parametric. In the spirit of this Note, we will use
maximum likelihood for estimation of β 1 and β 2 . We will begin with a fully parametric example.
Example 5.4 For the fire insurance example, consider the following ten payments. All values are
expressed as a percentage of the house’s value. Estimate the parameters of the Cox proportional
hazards model using maximum likelihood and both an exponential and a beta distribution for the
baseline hazard rate function. There is no deductible on these policies, but there is a policy limit
(which differs by policy).
z 1 z 2 payment
10 0 70
20 0 22
30 0 90*
40 0 81
50 0 8
10 1 51
20 1 95*
30 1 55
40 1 85*
50 1 93
An asterisk indicates that the payment was made at the policy limit.
In order to construct the likelihood function we need the density and survival functions. Let c j be
the Cox multiplier for the jth observation. Then, as noted in the previous example, S j (x) =S 0 (x) c j
,
where S 0 (x) is the baseline distribution. The density function is
For the exponential distribution,
f j (x) = −Sj(x) 0 =−c j S 0 (x) cj−1 S0(x)
0
= c j S 0 (x) cj−1 f 0 (x).
S j (x) =[e −x/θ ] c j
= e −c jx/θ and f j (x) =(c j /θ)e −c jx/θ