added definition "est-bias"

Author: JoramSoch

D/est-bias.md

@@ -0,0 +1,34 @@
+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2024-11-08 10:53:01
+
+title: "Biased vs. unbiased estimator"
+chapter: "General Theorems"
+section: "Estimation theory"
+topic: "Basic concepts of estimation"
+definition: "Biased vs. unbiased"
+
+sources:
+  - authors: "Wikipedia"
+    year: 2024
+    title: "Estimator"
+    in: "Wikipedia, the free encyclopedia"
+    pages: "retrieved on 2024-11-08"
+    url: "https://en.wikipedia.org/wiki/Estimator#Bias"
+
+def_id: "D209"
+shortcut: "est-bias"
+username: "JoramSoch"
+---
+
+
+**Definition:** Let $\hat{\theta}: \mathcal{Y} \rightarrow \Theta$ be an [estimator](/D/est) of a [parameter](/D/para) $\theta \in \Theta$ from [data](/D/data) $y \in \mathcal{Y}$. Then,
+
+* $\hat{\theta}$ is called an unbiased estimator when its [expected value](/D/mean) is equal to the parameter that it is estimating: $\mathrm{E}_{\hat{\theta}}\left[ \hat{\theta} \right] = \theta$, where the expectation is calculated over all possible samples $y$ leading to values of $\hat{\theta}$.
+
+* $\hat{\theta}$ is called a biased estimator otherwise, i.e. when $\mathrm{E}_{\hat{\theta}}\left[ \hat{\theta} \right] \neq \theta$.