Estimation, Evaluation, and Selection of Actuarial Models

125
x F (x) compare to max difference
0.1 0.174 0, 0.2 0.174
0.2 0.306 0.2, 0.4 0.106
0.5 0.556 0.4, 0.6 0.156
1.0 0.750 0.6, 0.8 0.150
1.3 0.811 0.8, 1 0.189
The test statistic is the maximum from the final column, 0.189.
Exercise 81 For Data Set B truncated at 50, the test statistic is 0.1631 which is less than the
critical value of 2.492 and the null hypothesis is not rejected. For Data Set B censored at 1,000,
the test statistic is 0.1712 and the null hypothesis is again not rejected.
Exercise 82 The calculations for Data Set B truncated at 50 are in the following table:
Range ˆp Expected Observed χ 2
50—150 0.1599 3.038 3 0.0005
150—250 0.1181 2.244 3 0.2545
250—500 0.2064 3.922 4 0.0015
500—1,000 0.2299 4.368 4 0.0310
1,000—2,000 0.1842 3.500 3 0.0713
2,000—∞ 0.1015 1.928 2 0.0027
The sum is 0.3615. With three degrees of freedom, the critical value is 7.8147 and the Weibull
model is not rejected. The p-value is 0.9481.
The calculations for Data Set B censored at 1,000 are in the following table:
Range ˆp Expected Observed χ 2
0—150 0.1886 3.772 4 0.0138
150—250 0.1055 2.110 3 0.3754
250—500 0.2076 4.151 4 0.0055
500—1,000 0.2500 4.999 4 0.1997
1,000—∞ 0.2484 4.968 5 0.0002
The sum is 0.5947. With two degrees of freedom, the critical value is 5.9915 and the Weibull model
is not rejected. The p-value is 0.7428.
The calculations for Data Set C are in the following table:
Range ˆp Expected Observed χ 2
7,500—17,500 0.3299 42.230 42 0.0013
17,500—32,500 0.2273 29.096 29 0.0003
32,500—67,500 0.2178 27.878 28 0.0005
67,500—125,000 0.1226 15.690 17 0.1094
125,000—300,000 0.0818 10.472 9 0.2070
300,000—∞ 0.0206 2.632 3 0.0513
The sum is 0.3698. With three degrees of freedom, the critical value is 7.8147 and the Weibull
model is not rejected. The p-value is 0.9464.
Exercise 83 The calculations are in the following table: