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 αⱼ, βⱼ, (j = 1,2, ⋯, n) be any numbers. Let Pₙ = P(αⱼ < Xⱼⁿ βⱼ, j = 1,2, ⋯, n). A recursion is given which calculates Pₙ for any F and any αⱼ, βⱼ. Suppose now that F is continuous. A two-sided statistic of Kolmogorov-Smirnov type has the distribution function PKS = P n1/2ψ(F) · |Fⁿ - F| λ, where Fⁿ is the empirical distribution function of the sample and ψ(x) is any nonnegative weight function. As PKS has the form Pₙ, its calculation as a function of λ can be carried out by means of the recursion. This has been done for the case ψ(x) = x(1 - x)-1/2. Curves are given which represent λ versus 1 - PKS for $n = 1,2, 10, 100$. From additional computations, the precision of a truncated development of 1 - PKS in powers of λ⁻² has been determined.