Limiting Distributions of Kolmogorov-Smirnov Type Statistics Under the Alternative

Let X₁, X₂, ⋯ be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let Y₁, Y₂, ⋯ be a sequence of i.i.d. variables with continuous cdfF(t) and with [0, 1] support. Let Fₙ(t, ω) denote the empirical distribution function based on Y₁(ω), ⋯, Yₙ(ω) and let Gₘ(t, ω) the empirical cdf pertaining to X₁(ω), ⋯, Xₘ(ω). Let 0 t 1|F(t) - t| = λ and Dₙ = 0 t 1|Fₙ(t, ω) - t|. The limiting distribution of n1/2(Dₙ - λ) is obtained in this paper. The limiting distributions under the alternative of the corresponding one-sided statistic in the one-sample case and the corresponding Smirnov statistics in the two-sample case are also derived. The asymptotic distributions under the alternative of Kuiper's statistic are also obtained.