Estimation, Evaluation, and Selection of Actuarial Models

4.5. HYPOTHESIS TESTS 75
4.5.2 Anderson-Darling test
This test is similar to the Kolmogorov-Smirnov test, but uses a different measure of the difference
between the two distribution functions. The test statistic is
Z u
A 2 [F n (x) − F ∗ (x)] 2
= n
t F ∗ (x)[1 − F ∗ (x)] f ∗ (x)dx.
That is, it is a weighted average of the squared differences between the empirical and model distribution
functions. Note that when x is close to t or to u, the weights might be very large due
to the small value of one of the factors in the denominator. This test statistic tends to place more
emphasis on good fit in the tails than in the middle of the distribution. Calculating with this
formula appears to be challenging. However, for individual data (so this is another test that does
not work for grouped data), the integral simplifies to
kX
A 2 = −nF ∗ (u)+n [1 − F n (y j )] 2 {ln[1 − F ∗ (y j )] − ln[1 − F ∗ (y j+1 )]}
j=0
kX
+n F n (y j ) 2 [ln F ∗ (y j+1 ) − ln F ∗ (y j )]
j=1
where the unique non-censored data points are t = y 0