{
  "id": "1706.01478",
  "version": "v1",
  "published": "2017-06-05T18:09:36.000Z",
  "updated": "2017-06-05T18:09:36.000Z",
  "title": "An optimal $(ε,δ)$-approximation scheme for the mean of random variables with bounded relative variance",
  "authors": [
    "Mark Huber"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "stat.CO"
  ],
  "abstract": "Randomized approximation algorithms for many #P-complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a $\\{0,1\\}$-matrix, and many others) reduce to creating random variables $X_1,X_2,\\ldots$ with finite mean $\\mu$ and standard deviation$\\sigma$ such that $\\mu$ is the solution for the problem input, and the relative standard deviation $|\\sigma/\\mu| \\leq c$ for known $c$. Under these circumstances, it is known that the number of samples from the $\\{X_i\\}$ needed to form an $(\\epsilon,\\delta)$-approximation $\\hat \\mu$ that satisfies $\\mathbb{P}(|\\hat \\mu - \\mu| > \\epsilon \\mu) \\leq \\delta$ is at least $(2-o(1))\\epsilon^{-2} c^2\\ln(1/\\delta)$. We present here an easy to implement $(\\epsilon,\\delta)$-approximation $\\hat \\mu$ that uses $(2+o(1))c^2\\epsilon^{-2}\\ln(1/\\delta)$ samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-05T18:09:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62K25",
      "65C05",
      "68W25"
    ],
    "keywords": [
      "random variables",
      "bounded relative variance",
      "approximation scheme",
      "randomized approximation algorithms",
      "extra conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}