Summary
Adequacy of the Smirnov approximation to P (D mn ≧c/mn), of the exact distribution of the two sample Kolmogorov-Smirnov criterion,D mn ,m<=n is examined. The main finding is that accuracy depends as much on the sample sizes,m andn, as on the ratio, ρ=m/n. The Smirnov approximation is good whenn is an integer multiple ofm, especiallyn=m, 2m; poor otherwise. This contrasts with the Smirnov approximation to P (D mn >c/mn), where this ordering is reversed, the cases forn=m, 2m being poor. The merit of continuity correction, of order ofn −1 in theD mn scale, is demonstrated for the Smirnov approximation, .1≦ρ, as well as for the Kolmogorov approximation, ρ<.1. As an argument for the optimum choice of ρ, the casen=m+1 is shown to have much to recommend it over the casen=m. Finally the usefulness of theχ 2 distribution with 2 degress of freedom is illustrated as an approximation to the Smirnov distribution.
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This study was in part supported by USPHS-NIH Grant number 2 DO4 AH 0167 07 PHT Development Project.
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Kim, P.J. The Smirnov distribution. Ann Inst Stat Math 28, 267–275 (1976). https://doi.org/10.1007/BF02504745
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DOI: https://doi.org/10.1007/BF02504745