Computing the cumulative distribution function of the Kolmogorov–Smirnov statistic

Drew, John H.; Glen, Andrew G.; Leemis, Lawrence M.

doi:10.1016/s0167-9473(99)00069-9

Cited by 74 publications

(30 citation statements)

References 3 publications

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“…As it is well-known [5,6], the problem of a sample fit can be solved with the use of non-parametric statistics of Kolmogorov-Smirnov test ) ( ) ( sup…”

Section: Self-consistent Stationary Levelmentioning

confidence: 99%

Numerical Algorithm for Self-consistent Stationary Level for Multidimensional Non-stationary Time-series

Kislitsyn

Козлова²,

Masherov³

et al. 2017

KIAM Prepr.

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“…As it is well-known [5,6], the problem of a sample fit can be solved with the use of non-parametric statistics of Kolmogorov-Smirnov test ) ( ) ( sup…”

Section: Self-consistent Stationary Levelmentioning

confidence: 99%

Numerical Algorithm for Self-consistent Stationary Level for Multidimensional Non-stationary Time-series

Kislitsyn

Козлова²,

Masherov³

et al. 2017

KIAM Prepr.

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“…In general, 3n/2 − 1 is the number of piecewise polynomials for 1/2n ≤ d < 1 where 3n/2 is the smallest integer greater than or equal to 3n/2. Ruben and Gambino (1982) computed the coefficients of the piecewise polynomials for n ≤ 10 and Drew, Glen, and Leemis (2000) using rational arithmetic in Maple (Maplesoft 2008) computed them for n ≤ 31. The computer storage requirements are immense.…”

Section: The Exact Formulamentioning

confidence: 99%

“…Since all calculations in this paper are performed in Mathematica, the Maple program of Drew, Glen, and Leemis (2000) was translated to Mathematica and is contained the file ExactDistribution.nb. Although the Mathematica code is large (over 1.5 million bytes), it is simply a list of rational arithmetic equations where d = t/n is used to find the correct equation and that equation is then evaluated to give the p value.…”

Section: The Exact Formulamentioning

confidence: 99%

“…The cumulative sampling distribution for the two-sided one sample K-S test is a piecewise polynomial that is different for each sample size n and the complexity rapidly grows with increasing n so it has not been generated let alone used for n > 31 (Ruben and Gambino 1982;Drew, Glen, and Leemis 2000). Consequently, the limiting distribution, various recursion formulae, and various approximations have been used to evaluate the cumulative sampling distribution.…”

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confidence: 99%

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Rational ArithmeticMathematicaFunctions to Evaluate the Two-Sided One Sample K-S Cumulative Sampling Distribution

Brown¹,

Harvey²

2008

J. Stat. Soft.

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One of the most widely used goodness-of-fit tests is the two-sided one sample KolmogorovSmirnov (K-S) test which has been implemented by many computer statistical software packages. To calculate a two-sided p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the two-sided one sample K-S test, this paper identifies an exact formula for sample sizes up to 31, six recursion formulae, and one matrix formula that can be used to calculate a p value. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each of these formulae is implemented in rational arithmetic. For the six recursion formulae and the matrix formula, computational experience for sample sizes up to 500 shows that computational times are increasing functions of both the sample size and the number of digits in the numerator and denominator integers of the rational number test statistic. The computational times of the seven formulae vary immensely but the Durbin recursion formula is almost always the fastest. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval half-width) and tables of calculated half-widths are presented for sample sizes up to 500. Using calculated half-widths as input, computational times for the fastest formula, the Durbin recursion formula, are given for sample sizes up to two thousand.

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“…In a hypothesis testing application, computing the test statistic d is relatively easier than evaluating the cumulative sampling distribution to determine the p value, P [D n ≥ d]. The cumulative sampling distribution is a piecewise polynomial that is different for each sample size n and whose complexity rapidly grows with increasing n so that it has not even been generated let alone used for n > 31 (see Ruben and Gambino (1982) and Drew, Glen, and Leemis (2000)). Consequently, the limiting distribution, various recursion formulae, and various approximations have been used to evaluate the cumulative sampling distribution.…”

Section: Introductionmentioning

confidence: 99%

Rational ArithmeticMathematicaFunctions to Evaluate the One-sided One-sample K-S Cumulative Sampling Distribution

Brown¹,

Harvey²

2007

J. Stat. Soft.

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One of the most widely used goodness-of-fit tests is the Kolmogorov-Smirnov (K-S) family of tests which have been implemented by many computer statistical software packages. To calculate a p value (evaluate the cumulative sampling distribution), these packages use various methods including recursion formulae, limiting distributions, and approximations of unknown accuracy developed over thirty years ago. Based on an extensive literature search for the one-sided one-sample K-S test, this paper identifies two direct formulae and five recursion formulae that can be used to calculate a p value and then develops two additional direct formulae and four iterative versions of the direct formulae for a total of thirteen formulae. To ensure accurate calculation by avoiding catastrophic cancelation and eliminating rounding error, each formula is implemented in rational arithmetic. Linear search is used to calculate the inverse of the cumulative sampling distribution (find the confidence interval bandwidth). Extensive tables of bandwidths are presented for sample sizes up to 2, 000. The results confirm the hypothesis that as the number of digits in the numerator and denominator integers of the rational number test statistic increases, the computation time also increases. In comparing the computational times of the thirteen formulae, the direct formulae are slightly faster than their iterative versions and much faster than all the recursion formulae. Computational times for the fastest formula are given for sample sizes up to fifty thousand.

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Computing the cumulative distribution function of the Kolmogorov–Smirnov statistic

Cited by 74 publications

References 3 publications

Numerical Algorithm for Self-consistent Stationary Level for Multidimensional Non-stationary Time-series

Numerical Algorithm for Self-consistent Stationary Level for Multidimensional Non-stationary Time-series

Rational ArithmeticMathematicaFunctions to Evaluate the Two-Sided One Sample K-S Cumulative Sampling Distribution

Rational ArithmeticMathematicaFunctions to Evaluate the One-sided One-sample K-S Cumulative Sampling Distribution

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