Estimation, Evaluation, and Selection of Actuarial Models

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70 CHAPTER 4. MODEL EVALUATION AND SELECTION Exponential Fit 0.15 0.1 D(x) 0.05 0 -0.05 -0.1 0 500 1000 1500 2000 2500 3000 3500 x Model vs. data D(x) plot for Data Set B truncated at 50 The lack of fit for this model is magnified in this plot. There is no corresponding plot for grouped data. For Data Set B censored at 1,000, the plot must again end at that value. It appears below. Exponential Fit 0.15 0.1 0.05 D(x) 0 -0.05 -0.1 -0.15 0 200 400 600 800 1000 x Model vs. data D(x) plot for Data Set B censored at 1,000 The lack of fit continues to be apparent. ¤ Exercise 76 Repeat the previous example for a Weibull model.
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.
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