The empirical cumulative distribution for n observations y_{i} is defined as E(x) = Σ _{i} (y_{i} < x). The two onesided KolmogorovSmirnov test statistics statistics are given by
where F(x) is the hypothesized distribution or another empirical distribution. The probability distributions of these two statistics, given that the null hypothesis of equality of distributions is true, does not depend on what the hyposthesized distribution is, as long as it is continuous. Knuth gives a detailed description of how to analyze the significance of this pair of statistics. Many people use max(D_{n}^{+}, D_{n}^{}) instead, but the distribution of this statistic is more difficult to deal with.
Note that when the underlying independent variable is cyclic as with day of the year or day of the week, then Kuiper's test is more appropriate. Numerical Recipes is again a good source of information on this.
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