Chi-Squared, Kolmogorov-Smirnov, and Anderson-Darling Tests
Compare the three main goodness-of-fit tests used for loss distribution validation on Exam STAM.
Chi-Squared Test
The chi-squared test groups data into c intervals and compares observed counts O_j to expected counts E_j. The test statistic is the sum of (O_j - E_j)^2 / E_j. Under H0, this follows a chi-squared distribution with c - 1 - p degrees of freedom, where p is the number of estimated parameters. Each expected count should be at least 5. The test is flexible but results depend on the grouping choice, which introduces some subjectivity.
KS and Anderson-Darling Tests
The Kolmogorov-Smirnov test uses the maximum absolute difference between the empirical CDF F_n(x) and the fitted CDF F(x). The KS statistic D = max|F_n(x) - F(x)| is compared to critical values. It works with ungrouped data but is most sensitive to differences near the median. The Anderson-Darling test statistic integrates the squared CDF difference weighted by 1/(F(x)(1-F(x))), giving extra weight to tail discrepancies. For insurance applications where tail fit matters, Anderson-Darling is often preferred. Exam STAM requires computing all three statistics from data.