{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X$ be a random sample whose law $P_\\theta$ depends on an unknown parameter $\\theta$ in a set $\\Theta$.\nA *pivotal quantity* is a function $g(X, \\theta)$ of the sample and the parameter with the following property.\nThe [[law|0QB913WM5E]] of the [[random variable|0DKRR22AEQ]] $g(X, \\theta)$ under $P_\\theta$ is the same for every $\\theta \\in \\Theta$.\n\n## Remarks\n\nExample: let $X_1, \\dots, X_n$ be i.i.d. $\\mathcal{N}(\\mu, \\sigma^2)$ with $\\theta = (\\mu, \\sigma^2)$ unknown, let $\\bar X$ be the sample mean and $S$ the sample standard deviation.\nThen $T = \\sqrt{n}(\\bar X - \\mu)/S$ has Student's t-distribution with $n - 1$ degrees of freedom for every $\\theta$, so $T$ is a pivotal quantity.\nNon-example: in the same model, $\\sqrt{n}(\\bar X - \\mu)$ is not pivotal, since its law $\\mathcal{N}(0, \\sigma^2)$ depends on $\\sigma^2$.\nA pivotal quantity need not be a statistic, since it may involve $\\theta$.\nIf $P_\\theta(a \\le g(X, \\theta) \\le b) = 1 - \\alpha$ for every $\\theta$, then the set $\\set{\\theta : a \\le g(X, \\theta) \\le b}$ is a [[confidence region|0K4NMMJBG2]] of level $1 - \\alpha$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "PXFYDT3M52",
    "kind": "definition",
    "msc": [
      "62F25",
      "62E15"
    ],
    "name": "Pivotal quantity",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Pivotal_quantity"
      }
    ],
    "tag": "TFCWC3N41W",
    "uses": [
      "0DKRR22AEQ",
      "0QB913WM5E"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/TFCWC3N41W/",
  "links": [
    {
      "field": "defines",
      "target": "PXFYDT3M52"
    },
    {
      "field": "uses",
      "target": "0DKRR22AEQ"
    },
    {
      "field": "uses",
      "target": "0QB913WM5E"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/TFCWC3N41W.md",
  "msc": [
    "62F25",
    "62E15"
  ],
  "name": "Pivotal quantity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/TFC/TFCWC3N41W.md",
  "status": null,
  "tag": "TFCWC3N41W",
  "uri": "m2://TFCWC3N41W",
  "url": "https://tamnd.github.io/math2/wiki/pivotal-quantity-definition/",
  "writing": {
    "tag": "TFCWC3N41W",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}