KolmogorovSmirnov 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 (GlivenkoCantelli
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 KolmogorovSmirnov (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 TwoStage DeltaCorrected
KolmogorovSmirnov 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 TwoStage DeltaCorrected KS Test.
Reference
D'Agostino,
R. and Stephens, M. (1986). GoodnessofFit Techniques. Marcel Dekker,
Inc.
