# estimator

Let ${X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}$ be samples (with observations ${X}_{i}={x}_{i}$) from a distribution^{} with probability density function $f(X\mid \theta )$, where $\theta $ is a real-valued unknown parameter (http://planetmath.org/StatisticalModel) in $f$. Consider $\theta $ as a random variable^{} and let $\tau (\theta )$ be its realization.

An *estimator ^{}* for $\theta $ is a statistic

^{}${\eta}_{\theta}={\eta}_{\theta}({X}_{1},{X}_{2},\mathrm{\dots},{X}_{n})$ that is used to, loosely speaking, estimate $\tau (\theta )$. Any value ${\eta}_{\theta}({X}_{1}={x}_{1},{X}_{2}={x}_{2},\mathrm{\dots},{X}_{n}={x}_{n})$ of ${\eta}_{\theta}$ is called an

*estimate*of $\tau (\theta )$.

Example.
Let ${X}_{1},{X}_{2},\mathrm{\dots},{X}_{n}$ be iid from a normal distribution^{} $N(\mu ,{\sigma}^{2})$. Here the two parameters are the mean $\mu $ and the variance^{} ${\sigma}^{2}$. The sample mean $\overline{X}$ is an estimator of $\mu $, while the sample variance ${s}^{2}$ is an estimator of ${\sigma}^{2}$. In addition, sample median, sample mode, sample trimmed mean are all estimators of $\mu $. The statistic defined by

$$\frac{1}{n-1}\sum _{i=1}^{n}{({X}_{i}-m)}^{2},$$ |

where $m$ is a sample median, is another estimator of ${\sigma}^{2}$.

Title | estimator |
---|---|

Canonical name | Estimator |

Date of creation | 2013-03-22 14:52:22 |

Last modified on | 2013-03-22 14:52:22 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 62A01 |

Defines | estimate |