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# estimator

Let $X_{1},X_{2},\ldots,X_{n}$ 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 in $f$. Consider $\theta$ as a random variable and let $\tau(\theta)$ be its realization.

An *estimator* for $\theta$ is a statistic $\eta_{{\theta}}=\eta_{{\theta}}(X_{1},X_{2},\ldots,X_{n})$ that is used to, loosely speaking, estimate $\tau(\theta)$. Any value $\eta_{{\theta}}(X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n})$ of $\eta_{{\theta}}$ is called an *estimate* of $\tau(\theta)$.

Example. Let $X_{1},X_{2},\ldots,X_{n}$ 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 $\sigma^{2}$.

## Mathematics Subject Classification

62A01*no label found*

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