DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE
The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if Fn is an empirical distribution function for variables i.i.d. with a distribution function F, and Kn is the Kolmogorov statistic View the MathML source, then there is a constant C such that for any M>0, Pr(Kn>M)≤Cexp(−2M2). Massart p...
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| Format: | Article |
| Language: | en_US |
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Elsevier B.V.
2012
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| Online Access: | http://hdl.handle.net/1721.1/70024 https://orcid.org/0000-0002-6195-4161 |
| _version_ | 1811074686107254784 |
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| author | Wei, Fan Dudley, Richard M. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Wei, Fan Dudley, Richard M. |
| author_sort | Wei, Fan |
| collection | MIT |
| description | The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if Fn is an empirical distribution function for variables i.i.d. with a distribution function F, and Kn is the Kolmogorov statistic View the MathML source, then there is a constant C such that for any M>0, Pr(Kn>M)≤Cexp(−2M2). Massart proved that one can take C=2 (DKWM inequality), which is sharp for F continuous. We consider the analogous Kolmogorov–Smirnov statistic for the two-sample case and show that for m=n, the DKW inequality holds for n≥n0 for some C depending on n0, with C=2 if and only if n0≥458.
The DKWM inequality fails for the three pairs (m,n) with 1≤m<n≤3. We found by computer search that the inequality always holds for n≥4 if 1≤m<n≤200, and further for n=2m if 101≤m≤300. We conjecture that the DKWM inequality holds for all pairs m≤n with the 457+3=460 exceptions mentioned. |
| first_indexed | 2024-09-23T09:53:44Z |
| format | Article |
| id | mit-1721.1/70024 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:53:44Z |
| publishDate | 2012 |
| publisher | Elsevier B.V. |
| record_format | dspace |
| spelling | mit-1721.1/700242022-09-30T17:32:45Z DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE Two-sample Dvoretzky–Kiefer–Wolfowitz inequalities Wei, Fan Dudley, Richard M. Massachusetts Institute of Technology. Department of Mathematics Dudley, Richard M. Wei, Fan Dudley, Richard M. The Dvoretzky–Kiefer–Wolfowitz (DKW) inequality says that if Fn is an empirical distribution function for variables i.i.d. with a distribution function F, and Kn is the Kolmogorov statistic View the MathML source, then there is a constant C such that for any M>0, Pr(Kn>M)≤Cexp(−2M2). Massart proved that one can take C=2 (DKWM inequality), which is sharp for F continuous. We consider the analogous Kolmogorov–Smirnov statistic for the two-sample case and show that for m=n, the DKW inequality holds for n≥n0 for some C depending on n0, with C=2 if and only if n0≥458. The DKWM inequality fails for the three pairs (m,n) with 1≤m<n≤3. We found by computer search that the inequality always holds for n≥4 if 1≤m<n≤200, and further for n=2m if 101≤m≤300. We conjecture that the DKWM inequality holds for all pairs m≤n with the 457+3=460 exceptions mentioned. 2012-04-13T17:15:34Z 2012-04-13T17:15:34Z 2011-11 2011-07 Article http://purl.org/eprint/type/JournalArticle 0167-7152 http://hdl.handle.net/1721.1/70024 Wei, Fan, and Richard M. Dudley. “Two-sample Dvoretzky–Kiefer–Wolfowitz Inequalities.” Statistics & Probability Letters 82.3 (2012): 636–644. Web. [Published in a shorter form]. https://orcid.org/0000-0002-6195-4161 en_US http://dx.doi.org/10.1016/j.spl.2011.11.012 Statistics and Probability Letters Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Elsevier B.V. Prof. Dudley |
| spellingShingle | Wei, Fan Dudley, Richard M. DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title | DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title_full | DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title_fullStr | DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title_full_unstemmed | DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title_short | DVORETZKY–KIEFER–WOLFOWITZ INEQUALITIES FOR THE TWO-SAMPLE CASE |
| title_sort | dvoretzky kiefer wolfowitz inequalities for the two sample case |
| url | http://hdl.handle.net/1721.1/70024 https://orcid.org/0000-0002-6195-4161 |
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