Abstract: The distribution of the Kolmogorov-Smirnov (K-S) test statistic has been widely studied under the assumption that the underlying theoretical cdf, F(x), is continuous. However, there are many real-life applications in which fitting discrete or mixed distributions is required. Nevertheless, due to inherent difficulties, the distribution of the K-S statistic when F(x) has jump discontinuities has been studied to a much lesser extent and no exact and efficient computational methods have been proposed in the literature.

In this talk, we will introduce a fast and accurate method to compute the (complementary) cdf of the K-S statistic when F(x) is discontinuous, and thus obtain exact p-values of the K-S test. The method works for small , medium and large sample sizes and is based on interpreting the K-S test as a double boundary crossing problem and applying the FFT method to compute the complimentary cdf. We give also an asymptotic formula for the distribution of the K-S statistics which generalizes the Schmidt’s asymptotic formula and thus allows for any mixed or purely discrete F(x). We also illustrate and discuss the numerical properties of both the exact and asymptotic methods in all cases, when F(x) is mixed, purely discrete, and continuous, based on examples from Finance and Insurance.

This is joint work with Dimitrina Dimitrova and Senren Tan.