Estimation, Evaluation, and Selection of Actuarial Models

68 CHAPTER 4. MODEL EVALUATION AND SELECTION
The fit is not as good as we might like because the model understates the distribution function
at smaller values of x and overstates the distribution function at larger values of x. This is not
good because it means that tail probabilities are understated.
For Data Set C, the likelihood function uses the truncated values. For example, the contribution
to the likelihood function for the first interval is
·F (17, 500) − F (7, 500)
1 − F (7, 500)
The maximum likelihood estimate is ˆθ =44, 253. The height of the first histogram bar is
¸42
.
42/[128(17, 500 − 7, 500)] = 0.0000328
and the last bar is for the interval from 125,000 to 300,000 (a bar cannot be constructed for the
interval from 300,000 to infinity). The density function must be truncated at 7,500 and becomes
f ∗ f(x)
(x) =
1 − F (7, 500) = 44, 253−1 e −x/44,253
1 − (1 − e −7,500/44,253 ) = e−(x−7,500)/44,253
, x>7, 500.
44, 253
The plot of the density function versus the histogram is given below.
Exponential Fit
0.000035
0.00003
0.000025
f(x)
0.00002
0.000015
0.00001
0.000005
0
0 50000 100000 150000 200000 250000 300000
x
Model
Empirical
Model vs. data density plot for Data Set C truncated at 7,500
The exponential model understates the early probabilities. It is hard to tell from the picture
how the curves compare above 125,000.
For Data Set B modified with a limit, the maximum likelihood estimate is ˆθ = 718.00. When
constructing the plot, the empirical distribution function must stop at 1,000. The plot appears
below.