{
  "aliases": [
    "method-of-moments-statistics"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nThe method of moments is a way to build [[estimators|0TTNRP5PFQ]] of unknown parameters.\nSuppose that the law of an i.i.d. sample $X_1, \\dots, X_n$ depends on $k$ real parameters $\\theta = (\\theta_1, \\dots, \\theta_k)$, and that the [[moments|0PH0VHTQS0]] $\\mu_j(\\theta) = E_\\theta[X_1^j]$ exist for $j \\le k$.\nThe method replaces each $\\mu_j(\\theta)$ with the sample moment $\\hat{m}_j = \\frac{1}{n} \\sum_{i=1}^{n} X_i^j$, and solves the $k$ equations $\\mu_j(\\hat{\\theta}) = \\hat{m}_j$ for $\\hat{\\theta}$.\n\n## Remarks\n\nExample: for $X_i \\sim \\operatorname{Exp}(\\lambda)$, $E[X_1] = 1/\\lambda$, so the method gives $\\hat{\\lambda} = 1/\\bar{X}_n$.\nExample: for $X_i \\sim \\mathcal{N}(\\mu, \\sigma^2)$, it gives $\\hat{\\mu} = \\bar{X}_n$ and $\\hat{\\sigma}^2 = \\frac{1}{n}\\sum_i (X_i - \\bar{X}_n)^2$.\nIf the map $\\theta \\mapsto (\\mu_1(\\theta), \\dots, \\mu_k(\\theta))$ has a continuous inverse, the law of large numbers makes the estimators [[consistent|0XJVF81QF6]].\nThe estimators are often simple, but they are usually less efficient than maximum likelihood estimators.\n\n\n\n## Explanations\n\n### L1 human\n\nThe method of moments chooses the parameters so that the model has the same averages as the data.\nFor example, if the data have mean $2$, an exponential model with mean $1/\\lambda$ gets the rate $\\lambda = 1/2$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "method-of-moments-statistics"
    ],
    "kind": "concept",
    "msc": [
      "62F10"
    ],
    "name": "Method of moments",
    "same_as": {
      "wikidata": "Q1925212",
      "wikipedia": "Method of moments (statistics)"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Method_of_moments_(statistics)"
      }
    ],
    "tag": "09J1H07VAH",
    "uses": [
      "0PH0VHTQS0",
      "0TTNRP5PFQ",
      "0QNRBHWV35",
      "0XJVF81QF6"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/09J1H07VAH/",
  "links": [
    {
      "field": "uses",
      "target": "0PH0VHTQS0"
    },
    {
      "field": "uses",
      "target": "0TTNRP5PFQ"
    },
    {
      "field": "uses",
      "target": "0QNRBHWV35"
    },
    {
      "field": "uses",
      "target": "0XJVF81QF6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/09J1H07VAH.md",
  "msc": [
    "62F10"
  ],
  "name": "Method of moments",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/09J/09J1H07VAH.md",
  "status": null,
  "tag": "09J1H07VAH",
  "uri": "m2://09J1H07VAH",
  "url": "https://math2.tamnd.com/wiki/method-of-moments/",
  "writing": {
    "tag": "09J1H07VAH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}