{
  "id": "1802.08883",
  "version": "v1",
  "published": "2018-02-24T18:13:48.000Z",
  "updated": "2018-02-24T18:13:48.000Z",
  "title": "Approximation of Kolmogorov-Smirnov Test Statistics",
  "authors": [
    "Long Bai",
    "David Kalaj"
  ],
  "comment": "27 page",
  "categories": [
    "math.PR"
  ],
  "abstract": "Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in this contribution, we concern the asymptotics of \\begin{align*} \\mathbb{P}\\left\\{\\sup_{\\boldsymbol{x}\\in [0,1]^n}\\left(W(\\boldsymbol{x})\\Big| W(\\boldsymbol{1})=w\\right)>u\\right\\}, \\ w\\in\\mathbb{R}, \\end{align*} for large $u$ where $W(\\boldsymbol{x})$ is the multivariate Brownian sheet based on a distribution function $F$. The results related to general $F$ are investigated and some important examples are also showed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-24T18:13:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70"
    ],
    "keywords": [
      "kolmogorov-smirnov test statistics",
      "approximation",
      "multivariate brownian sheet",
      "important examples",
      "distribution function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}