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### Kolmogorov-Smirnov Test

Numbers in parentheses correspond to the numbered references in my publication list.

In the 1930s, Kolmogorov and Smirnov developed a goodness of fit test for continuous data to determine if a sample comes from a given hypothesized distribution. Today it continues to be one of the best known and most widely used goodness of fit tests because of its simplicity and because it is based on the empirical distribution function (edf), which converges uniformly to the population cumulative distribution function (cdf) with probability measure one (Glivenko-Cantelli theorem). Even though a plethora of goodness of fit tests have been developed in recent decades (see, e.g., (32) and D'Agostino and Stephens 1986), many with higher statistical power than the Kolmogorov-Smirnov (KS) test, the KS test remains popular because it is simple and intuitive, comparing the edf to the cdf and basing the test on the maximum deviation.

In a series of articles ((3), (20), (28), (32), (48), and (54)) it has been determined that with a minor adjustment in the definition of the edf for the KS test procedure, the statistical power of the KS test can be significantly enhanced. In fact, the modified test, called the Two-Stage Delta-Corrected Kolmogorov-Smirnov Test, is uniformly at least as powerful as the classical KS test, with power improvements of up to 46 percentage points (54). Furthermore, the size (alpha) of the new test is not disturbed by the modification, and the conduct of the new test is no more complicated (i.e., it involves no more steps) than the classical KS test. A table of critical values and examples for the new test are given in (54).

Therefore, everywhere that the classical KS test is presented or used, it should be replaced with the Two-Stage Delta-Corrected KS Test.

#### Reference

D'Agostino, R. and Stephens, M. (1986). Goodness-of-Fit Techniques. Marcel Dekker, Inc.

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 Prof. Harry Khamis, Director [e-mail] Statistical Consulting Center [home][e-mail] 130 Mathematics and Microbiological Sciences Building Wright State University 3640 Colonel Glenn Highway Dayton OH 45435 USA office phone: (937) 775-2433 or stat consulting center (937) 775-4205 fax: (937) 775-2081 Last updated: July 18, 2001 All Wright State University Webpages must publish this disclaimer. Please send your comments, questions and suggestions regarding this webpage to me. (sjm)