{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be [[i.i.d.|0AJN7WHBJZ]] random vectors in $\\R^d$, and let $\\mu \\in \\R^d$.\nThe *empirical likelihood ratio* for the mean $\\mu$ is\n$$\\mathcal{R}(\\mu) = \\max\\Big\\{ \\prod_{i=1}^{n} n w_i \\;:\\; w_i \\ge 0, \\ \\sum_{i=1}^{n} w_i = 1, \\ \\sum_{i=1}^{n} w_i X_i = \\mu \\Big\\},$$\nwith $\\mathcal{R}(\\mu) = 0$ when no weights satisfy the constraints.\nMore generally, for a parameter $\\theta$ defined by $E[m(X, \\theta)] = 0$, the constraint $\\sum_i w_i X_i = \\mu$ is replaced by $\\sum_i w_i m(X_i, \\theta) = 0$.\n\n## Remarks\n\nThe factor $n$ makes the unconstrained maximum equal to $1$: without the mean constraint, $\\prod w_i$ is largest at $w_i = 1/n$.\nFeasible weights exist exactly when $\\mu$ lies in the [[convex hull|0TCHAS04WK]] of $X_1, \\dots, X_n$.\nWhen $\\mu$ is in the interior of the hull, the maximiser is $w_i = \\frac{1}{n(1 + \\lambda^{T}(X_i - \\mu))}$, where $\\lambda \\in \\R^d$ solves $\\sum_i \\frac{X_i - \\mu}{1 + \\lambda^{T}(X_i - \\mu)} = 0$.\nOwen's theorem: if $\\mu_0 = E[X_1]$ and the covariance matrix of $X_1$ is finite and nonsingular, then $-2 \\ln \\mathcal{R}(\\mu_0)$ converges in distribution to the [[chi-squared distribution|0RFMMBHFTV]] with $d$ degrees of freedom.\nSo $\\set{\\mu : -2\\ln\\mathcal{R}(\\mu) \\le c}$, with $c$ the $(1-\\alpha)$-quantile of $\\chi^2_d$, is an asymptotic confidence region of level $1 - \\alpha$.\n\nExample: $\\mu = \\bar X$.\nThe weights $w_i = 1/n$ satisfy the constraint, so $\\mathcal{R}(\\bar X) = 1$, the largest possible value.\n\nNon-example: data $(1, 2, 3)$ in $\\R$ and $\\mu = 5$.\nNo weighted average of $1, 2, 3$ equals $5$, because $5$ is outside the convex hull $[1, 3]$, so $\\mathcal{R}(5) = 0$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "BFQBHBN5F5",
    "kind": "definition",
    "msc": [
      "62G10",
      "62G15",
      "62G20"
    ],
    "name": "Empirical likelihood",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Empirical_likelihood"
      }
    ],
    "tag": "H4DD38RKBA",
    "uses": [
      "0AJN7WHBJZ",
      "05FSTXM92W",
      "0TCHAS04WK",
      "0RFMMBHFTV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/H4DD38RKBA/",
  "links": [
    {
      "field": "defines",
      "target": "BFQBHBN5F5"
    },
    {
      "field": "uses",
      "target": "0AJN7WHBJZ"
    },
    {
      "field": "uses",
      "target": "05FSTXM92W"
    },
    {
      "field": "uses",
      "target": "0TCHAS04WK"
    },
    {
      "field": "uses",
      "target": "0RFMMBHFTV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/H4DD38RKBA.md",
  "msc": [
    "62G10",
    "62G15",
    "62G20"
  ],
  "name": "Empirical likelihood",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/H4D/H4DD38RKBA.md",
  "status": null,
  "tag": "H4DD38RKBA",
  "uri": "m2://H4DD38RKBA",
  "url": "https://tamnd.github.io/math2/wiki/empirical-likelihood-definition/",
  "writing": {
    "tag": "H4DD38RKBA",
    "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
      }
    ]
  }
}