A The Power of Alternative Kolmogorov-Smirnov Tests Based on Transformations of the Data

The Kolmogorov-Smirnov (KS) statistical test is commonly used to determine if data can be regarded as a sample from a sequence of i.i.d. random variables with specified continuous cdf F , but with small samples it can have insufficient power, i.e., its probability of rejecting natural alternatives can be too low. However, Durbin [1961] showed that the power of the KS test often can be increased, for given significance level, by a well-chosen transformation of the data. Simulation experiments reported here show that the power can often be more consistently and substantially increased by modifying the original Durbin transformation by first transforming the given sequence to a sequence of mean-1 exponential random variables, which is equivalent to a rate-1 Poisson process, and then applying the classical conditional-uniform transformation to convert the arrival times into the order statistics of i.i.d. uniform random variables. The new KS test often has much more power, because it focuses on the cumulative sums rather than the random variables themselves.