{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $p$ and $m$ be positive integers with $m \\ge p$, and let $\\Sigma$ be a $p \\times p$ [[positive definite matrix|0QQCG3JWJ4]].\nLet $x$ be a random vector with the [[multivariate normal distribution|0G558GJQMW]] $\\mathcal{N}_p(0, \\Sigma)$.\nLet $M$ be a random matrix with the [[Wishart distribution|8BEMNJW0B9]] $W_p(\\Sigma, m)$, [[independent|0E9502NT92]] of $x$.\nThe distribution of\n$$T^2 = m \\, x^T M^{-1} x$$\ndoes not depend on $\\Sigma$.\nIt is *Hotelling's T-squared distribution* with dimension $p$ and $m$ degrees of freedom, written $T^2(p, m)$.\n\n## Remarks\n\nThe condition $m \\ge p$ makes $M$ invertible with probability $1$.\nThe distribution is a rescaled [[F-distribution|PBT8VRJADK]]:\n$$\\frac{m - p + 1}{p m} T^2 \\sim F(p, m - p + 1).$$\n\nExample: for $p = 1$, $x = \\sigma Z$ with $Z$ standard normal and $M = \\sigma^2 V$ with $V$ chi-squared with $m$ degrees of freedom.\nThen $T^2 = Z^2/(V/m)$ is the square of a [[Student t|0NQ5798SB1]] variable with $m$ degrees of freedom, and the formula above gives $T^2 \\sim F(1, m)$.\n\nExample: let $Y_1, \\dots, Y_n$ be i.i.d. $\\mathcal{N}_p(\\mu, \\Sigma)$ with $n - 1 \\ge p$, with sample mean $\\bar{Y}$ and sample covariance matrix $\\hat{\\Sigma}$.\nThen $\\sqrt{n}(\\bar{Y} - \\mu) \\sim \\mathcal{N}_p(0, \\Sigma)$ and $(n - 1)\\hat{\\Sigma} \\sim W_p(\\Sigma, n - 1)$ are independent, so $n (\\bar{Y} - \\mu)^T \\hat{\\Sigma}^{-1} (\\bar{Y} - \\mu) \\sim T^2(p, n - 1)$.\n\nNon-example: if $M$ and $x$ are dependent, for instance $p = 1$, $m = 1$ and $M = x^2$, then $T^2 = 1$ always, which is not $T^2(1, 1)$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XS9GETCZX2",
    "kind": "definition",
    "msc": [
      "62H10",
      "62H15"
    ],
    "name": "Hotelling's T-squared distribution",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hotelling%27s_T-squared_distribution"
      }
    ],
    "tag": "SR8DQDKHQN",
    "uses": [
      "0G558GJQMW",
      "8BEMNJW0B9",
      "0E9502NT92",
      "0QQCG3JWJ4",
      "PBT8VRJADK",
      "0NQ5798SB1"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/SR8DQDKHQN/",
  "links": [
    {
      "field": "defines",
      "target": "XS9GETCZX2"
    },
    {
      "field": "uses",
      "target": "0G558GJQMW"
    },
    {
      "field": "uses",
      "target": "8BEMNJW0B9"
    },
    {
      "field": "uses",
      "target": "0E9502NT92"
    },
    {
      "field": "uses",
      "target": "0QQCG3JWJ4"
    },
    {
      "field": "uses",
      "target": "PBT8VRJADK"
    },
    {
      "field": "uses",
      "target": "0NQ5798SB1"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/SR8DQDKHQN.md",
  "msc": [
    "62H10",
    "62H15"
  ],
  "name": "Hotelling's T-squared distribution",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SR8/SR8DQDKHQN.md",
  "status": null,
  "tag": "SR8DQDKHQN",
  "uri": "m2://SR8DQDKHQN",
  "url": "https://tamnd.github.io/math2/wiki/hotellings-t-squared-distribution-definition/",
  "writing": {
    "tag": "SR8DQDKHQN",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}