# empirical distribution function

Let ${X}_{1},\mathrm{\dots},{X}_{n}$ be random variables^{} with realizations ${x}_{i}={X}_{i}(\omega )\in \mathbb{R}$, $i=1,\mathrm{\dots},n$. The *empirical distribution function* ${F}_{n}(x,\omega )$ based on ${x}_{1},\mathrm{\dots},{x}_{n}$ is

$${F}_{n}(x,\omega )=\frac{1}{n}\sum _{i=1}^{n}{\chi}_{{A}_{i}}(x,\omega ),$$ |

where ${\chi}_{{A}_{i}}$ is the indicator function^{} (or characteristic function^{}) and ${A}_{i}=\{(x,\omega )\mid {x}_{i}\le x\}$. Note that each indicator function is itself a random variable.

The empirical function can be alternatively and equivalently defined by using the order statistics^{} ${X}_{(i)}$ of ${X}_{i}$ as:

$$ |

where ${x}_{(i)}$ is the realization of the random variable ${X}_{(i)}$ with outcome $\omega $.

Title | empirical distribution function |
---|---|

Canonical name | EmpiricalDistributionFunction |

Date of creation | 2013-03-22 14:33:27 |

Last modified on | 2013-03-22 14:33:27 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 7 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 62G30 |