{
  "id": "1305.5429",
  "version": "v2",
  "published": "2013-05-23T14:16:35.000Z",
  "updated": "2013-05-24T17:39:03.000Z",
  "title": "Gaussian Mills ratio is completely monotone",
  "authors": [
    "Armengol Gasull",
    "Frederic Utzet"
  ],
  "comment": "This paper has been withdrawn by the authors due that the results appear in Arpad Baricz. Mills'ratio: Monotonicity patterns and functional inequalities. Journal of Mathematical Analysis and Applications, 340 (2008) 1362-1370 M. R. Sampford. Inequalities on Mill's ratio and related functions, The Annals of Mathematical Statistics, Vol. 24, No. 1 (1953) 130-132",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the Mills ratio corresponding to the standard Gaussian law, $f(x)=\\big(1-\\Phi(x)\\big)/\\phi(x), \\, x\\ge 0$, where $\\phi$ is the density function of this law and $\\Phi$ its cumulative distribution function. We prove that this function is completely monotone. In the proof we obtain a sequence of rational functions that are sharp bounds for $f$; it turns out that these rational functions are the convergents of the continued fraction defined by $f$, and provide an approximation procedure that allows to prove interesting properties where $f$ or its derivatives are involved. As an application we show that $1/f$ is strictly convex.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-24T17:39:03.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gaussian mills ratio",
      "rational functions",
      "standard gaussian law",
      "density function",
      "approximation procedure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.5429G"
    }
  }
}