Computing the Kolmogorov-Smirnov Distribution When the Underlying CDF is Purely Discrete, Mixed, or Continuous
Dublin Core
Title
Computing the Kolmogorov-Smirnov Distribution When the Underlying CDF is Purely Discrete, Mixed, or Continuous
Subject
Kolmogorov-Smirnov test statistic, discontinuous (discrete or mixed) distribution,
fast Fourier transform, double boundary non-crossing, rectangle probability for uniform order
statistics
fast Fourier transform, double boundary non-crossing, rectangle probability for uniform order
statistics
Description
The distribution of the Kolmogorov-Smirnov (KS) test statistic has been widely studied under the assumption that the underlying theoretical cumulative distribution function
(CDF), F(x), is continuous. However, there are many real-life applications in which fitting
discrete or mixed distributions is required. Nevertheless, due to inherent difficulties, the
distribution of the KS statistic when F(x) has jump discontinuities has been studied to a
much lesser extent and no exact and efficient computational methods have been proposed
in the literature.
In this paper, we provide a fast and accurate method to compute the (complementary)
CDF of the KS statistic when F(x) is discontinuous, and thus obtain exact p values of
the KS test. Our approach is to express the complementary CDF through the rectangle
probability for uniform order statistics, and to compute it using fast Fourier transform
(FFT). Secondly, we provide a C++ and an R implementation of the proposed method,
which fills the existing gap in statistical software. We give also a useful extension of
the Schmid’s asymptotic formula for the distribution of the KS statistic, relaxing his
requirement for F(x) to be increasing between jumps and thus allowing for any general
mixed or purely discrete F(x). The numerical performance of the proposed FFT-based
method, implemented both in C++ and in the R package KSgeneral, available from
https://CRAN.R-project.org/package=KSgeneral, is illustrated when F(x) is mixed,
purely discrete, and continuous. The performance of the general asymptotic formula is
also studied
(CDF), F(x), is continuous. However, there are many real-life applications in which fitting
discrete or mixed distributions is required. Nevertheless, due to inherent difficulties, the
distribution of the KS statistic when F(x) has jump discontinuities has been studied to a
much lesser extent and no exact and efficient computational methods have been proposed
in the literature.
In this paper, we provide a fast and accurate method to compute the (complementary)
CDF of the KS statistic when F(x) is discontinuous, and thus obtain exact p values of
the KS test. Our approach is to express the complementary CDF through the rectangle
probability for uniform order statistics, and to compute it using fast Fourier transform
(FFT). Secondly, we provide a C++ and an R implementation of the proposed method,
which fills the existing gap in statistical software. We give also a useful extension of
the Schmid’s asymptotic formula for the distribution of the KS statistic, relaxing his
requirement for F(x) to be increasing between jumps and thus allowing for any general
mixed or purely discrete F(x). The numerical performance of the proposed FFT-based
method, implemented both in C++ and in the R package KSgeneral, available from
https://CRAN.R-project.org/package=KSgeneral, is illustrated when F(x) is mixed,
purely discrete, and continuous. The performance of the general asymptotic formula is
also studied
Creator
Dimitrina S. Dimitrova
Source
https://www.jstatsoft.org/article/view/v095i10
Publisher
City, University of London
Date
October 2020
Contributor
Fajar bagus W
Format
PDF
Language
Inggris
Type
Text
Files
Collection
Citation
Dimitrina S. Dimitrova, “Computing the Kolmogorov-Smirnov Distribution When the Underlying CDF is Purely Discrete, Mixed, or Continuous,” Repository Horizon University Indonesia, accessed April 4, 2025, https://repository.horizon.ac.id/items/show/8160.