{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "09J1H07VAH"
    },
    {
      "field": "uses",
      "target": "0FREPFKNS7"
    }
  ],
  "body": "\n## Definition\n\nFor each $n \\ge 1$, let $\\hat{\\theta}_n$ be an [[estimator|0TTNRP5PFQ]] of a real parameter $\\theta$ that uses the first $n$ observations of a sample.\nThe sequence $(\\hat{\\theta}_n)$ is consistent if, for each $\\theta \\in \\Theta$, $\\hat{\\theta}_n \\to \\theta$ [[in probability|0EFDDXTVD6]] when $\\theta$ is the true parameter:\n$$\\lim_{n \\to \\infty} P_\\theta\\left(|\\hat{\\theta}_n - \\theta| \u003e \\varepsilon\\right) = 0 \\quad \\text{for each } \\varepsilon \u003e 0.$$\n\n## Remarks\n\nIf the convergence is almost sure, the sequence is strongly consistent.\nBy the weak law of large numbers, the sample mean of an i.i.d. sample is a consistent estimator of the mean.\nIf the mean squared error of $\\hat{\\theta}_n$ tends to $0$, then $(\\hat{\\theta}_n)$ is consistent. This follows from Markov's inequality.\nConsistent estimators can be biased, and unbiased estimators can fail to be consistent. For example, the estimator $X_1$ of the mean uses only one observation: it is unbiased but not consistent, unless $X_1$ is almost surely constant.\n\n## Explanations\n\n### L1 human\n\nA consistent estimator gets closer and closer to the truth when you collect more and more data.\nThis is a minimal requirement for a reasonable estimator.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0AS800VAYX",
    "kind": "definition",
    "msc": [
      "62F12"
    ],
    "name": "Consistent estimator",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Consistent_estimator"
      }
    ],
    "tag": "0XJVF81QF6",
    "uses": [
      "0TTNRP5PFQ",
      "0EFDDXTVD6"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0XJVF81QF6/",
  "links": [
    {
      "field": "defines",
      "target": "0AS800VAYX"
    },
    {
      "field": "uses",
      "target": "0TTNRP5PFQ"
    },
    {
      "field": "uses",
      "target": "0EFDDXTVD6"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0XJVF81QF6.md",
  "msc": [
    "62F12"
  ],
  "name": "Consistent estimator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0XJ/0XJVF81QF6.md",
  "status": null,
  "tag": "0XJVF81QF6",
  "uri": "m2://0XJVF81QF6",
  "url": "https://math2.tamnd.com/wiki/consistent-estimator-definition/",
  "writing": {
    "tag": "0XJVF81QF6",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}