Editing Estimator (section)

===Unbiased===
[[File:Wiki Snipet Unbiased.png|right|thumb|upright|Difference between estimators: an unbiased estimator <math>\theta_2</math> is centered around <math>\theta</math> vs. a biased estimator <math>\theta_1</math>.]]

A desired property for estimators is the unbiased trait where an estimator is shown to have no systematic tendency to produce estimates larger or smaller than the true parameter. Additionally, unbiased estimators with smaller variances are preferred over larger variances because it will be closer to the "true" value of the parameter. The unbiased estimator with the smallest variance is known as the [[minimum-variance unbiased estimator]] (MVUE).

To find if your estimator is unbiased it is easy to follow along the equation <math>\operatorname E(\widehat{\theta}) - \theta=0</math>, <math>\widehat{\theta}</math>. With estimator ''T'' with and parameter of interest <math>\theta</math> solving the previous equation so it is shown as <math>\operatorname E[T] = \theta</math> the estimator is unbiased. Looking at the figure to the right despite <math>\hat{\theta_2}</math> being the only unbiased estimator, if the distributions overlapped and were both centered around <math>\theta</math> then distribution <math>\hat{\theta_1}</math> would actually be the preferred unbiased estimator.

'''Expectation'''
When looking at quantities in the interest of expectation for the model distribution there is an unbiased estimator which should satisfy the two equations below.
:<math>1. \quad \overline X_n = \frac{X_1 + X_2+ \cdots + X_n} n</math>
:<math>2. \quad \operatorname E\left[\overline X_n \right] = \mu</math>
'''Variance'''
Similarly, when looking at quantities in the interest of variance as the model distribution there is also an unbiased estimator that should satisfy the two equations below.
:<math>1. \quad S^2_n = \frac{1}{n-1}\sum_{i = 1}^n (X_i - \bar{X_n})^2</math>
:<math>  2. \quad \operatorname E\left[S^2_n\right] = \sigma^2</math>
Note we are dividing by ''n''&nbsp;−&nbsp;1 because if we divided with ''n'' we would obtain an estimator with a negative bias which would thus produce estimates that are too small for <math>\sigma^2</math>. It should also be mentioned that even though <math>S^2_n</math> is unbiased for <math>\sigma^2</math> the reverse is not true.<ref name=Dekker2005>{{Cite book|last1=Dekking|first1=Frederik Michel|last2=Kraaikamp|first2=Cornelis|last3=Lopuhaä|first3=Hendrik Paul|last4=Meester|first4=Ludolf Erwin|date=2005|title=A Modern Introduction to Probability and Statistics|url=https://archive.org/details/modernintroducti0000unse_h6a1 | series=Springer Texts in Statistics|language=en-gb|isbn=978-1-85233-896-1}}</ref>