The Calculation of Distributions of Two-Sided Kolmogorov-Smirnov Type Statistics

Let X₁ⁿ X₂ⁿ Xₙⁿ be the order statistics of a size n sample from any distribution function F not necessarily continuous. Let ⱼ, ⱼ, (j = 1, 2, , n) be any numbers. Let Pₙ = P (ⱼ < Xⱼⁿ ⱼ, j = 1, 2, , n). A recursion is given which calculates Pₙ for any F and any ⱼ, ⱼ. Suppose now that F is continuous. A two-sided statistic of Kolmogorov-Smirnov type has the distribution function P₊ₒ = P n^1{2} (F) |Fⁿ - F|, where Fⁿ is the empirical distribution function of the sample and (x) is any nonnegative weight function. As P₊ₒ has the form Pₙ, its calculation as a function of can be carried out by means of the recursion. This has been done for the case (x) = x (1 - x) ^-1{2}. Curves are given which represent versus 1 - P₊ₒ for n = 1, 2, 10, 100. From additional computations, the precision of a truncated development of 1 - P₊ₒ in powers of ^-2 has been determined.