{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "B7HPB2Z6GC"
    },
    {
      "field": "uses",
      "target": "H5G7CNYZEB"
    },
    {
      "field": "uses",
      "target": "KZDMD3SDJA"
    },
    {
      "field": "uses",
      "target": "N2TSHRCJ1Y"
    },
    {
      "field": "uses",
      "target": "SARNZPAKE5"
    },
    {
      "field": "uses",
      "target": "W94W93RHF5"
    },
    {
      "field": "about",
      "target": "H5G7CNYZEB"
    },
    {
      "field": "defines",
      "target": "N1NA5MEFWX"
    }
  ],
  "body": "\n## Summary\n\nThe empirical distribution function of a sample is the [[cumulative distribution function|0H8H7ZTHW2]] of the distribution that puts mass $1/n$ on each observation.\nAt each point $t$ it gives the fraction of observations that are at most $t$.\nIt estimates the true distribution function without any parametric assumption, and for an i.i.d. sample the [[Glivenko–Cantelli theorem|E7B78X780X]] shows that it converges to it uniformly, almost surely.\nGoodness-of-fit tests such as the [[Kolmogorov–Smirnov test|YZYHWC6C5F]] compare it with a hypothesised distribution function.\n\n## Explanations\n\n### L1 human\n\nSort your data and draw a staircase that starts at height $0$ and jumps up by $1/n$ at each data point.\nThis staircase is the empirical distribution function.\nFor the data $1, 2, 2, 5$, it is $0$ before $1$, $1/4$ on $[1, 2)$, $3/4$ on $[2, 5)$ and $1$ from $5$ on.\nWith more and more data, the staircase hugs the true distribution function more and more closely.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "62G30"
    ],
    "name": "Empirical distribution function",
    "same_as": {
      "wikidata": "Q1339385",
      "wikipedia": "Empirical distribution function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Empirical_distribution_function"
      }
    ],
    "tag": "A05WR6TEK5"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/A05WR6TEK5/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/A05WR6TEK5.md",
  "msc": [
    "62G30"
  ],
  "name": "Empirical distribution function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A05/A05WR6TEK5.md",
  "status": null,
  "tag": "A05WR6TEK5",
  "uri": "m2://A05WR6TEK5",
  "url": "https://tamnd.github.io/math2/wiki/empirical-distribution-function/",
  "writing": {
    "tag": "A05WR6TEK5",
    "kind": "concept",
    "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
      }
    ]
  }
}