{
  "id": "1401.2326",
  "version": "v3",
  "published": "2014-01-10T13:33:16.000Z",
  "updated": "2014-08-05T09:05:59.000Z",
  "title": "Inference in $α$-Brownian bridge based on Karhunen-Loève expansions",
  "authors": [
    "Maik Görgens"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a simple decision problem on the scaling parameter in the $\\alpha$-Brownian bridge $X^{(\\alpha)}$ on the interval $[0,1]$: given two values $\\alpha_0, \\alpha_1 \\geq 0$ with $\\alpha_0 + \\alpha_1 \\geq 1$ and some time $0 \\leq T \\leq 1$ we want to test $H_0: \\alpha = \\alpha_0$ vs. $H_1: \\alpha = \\alpha_1$ based on the observation of $X^{(\\alpha)}$ until time $T$. The likelihood ratio can be written as a functional of a quadratic form $\\psi(X^{(\\alpha)})$ of $X^{(\\alpha)}$. In order to calculate the distribution of $\\psi(X^{(\\alpha)})$ under the null hypothesis, we generalize the Karhunen-Lo\\`eve Theorem to positive finite measures on $[0,1]$ and compute the Karhunen-Lo\\`eve expansion of $X^{(\\alpha)}$ under such a measure. Based on this expansion, the distribution of $\\psi(X^{(\\alpha)})$ follows by Smirnov's formula.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-08-05T09:05:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "62M02"
    ],
    "keywords": [
      "brownian bridge",
      "karhunen-loève expansions",
      "simple decision problem",
      "likelihood ratio",
      "smirnovs formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.2326G"
    }
  }
}