Estimation, Evaluation, and Selection of Actuarial Models

4.4. GRAPHICAL COMPARISON OF THE DENSITY AND DISTRIBUTION FUNCTIONS71
Another popular way to highlight any differences is the p−p plot which is also called a probability
plot. The plot is created by ordering the observations as x 1 ≤ ··· ≤ x n . A point is then plotted
corresponding to each value. The coordinates to plot are (F n (x j ),F ∗ (x j )). 3 If the model fits well,
the plotted points will be near the forty-five degree line. However, for this to be the case a different
definition of the empirical distribution function is needed. It can be shown that the expected value
of F n (x j ) is j/(n +1) and therefore the empirical distribution should be that value and not the
usual j/n. If two observations have the same value, either plot both points (they would have the
same “y” value, but different “x” values) or plot a single value by averaging two “x” values.
Example 4.13 Create a p − p plot for the continuing example.
For Data Set B truncated at 50, n =19and one of the observed values is x =82. The empirical
value is F n (82) = 1/20 = 0.05. The other coordinate is
F ∗ (82) = 1 − e −(82−50)/802.32 =0.0391.
One of the plotted points will be (0.05, 0.0391). The complete picture appears below.
Exponential Fit
F(x)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1
F n (x)
p − p plot for Data Set B truncated at 50
Fromthelowerleftpartoftheplotitisclearthattheexponentialmodelplaceslessprobability
on small values than the data calls for. A similar plot can be constructed for Data Set B censored
at 1,000.
3 In the text Loss Models this plot was incorrectly called a q − q plot. There is a plot that goes by that name, but
it will not be introduced in this Note.