View source for Biased estimator

A statistical estimator whose expectation does not coincide with the value being estimated.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160501.png" /> be a random variable taking values in a sampling space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160503.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160504.png" /> be a statistical point estimator of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160505.png" /> defined on the parameter set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160506.png" />. It is assumed that the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160507.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160508.png" /> exists. If the function

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b0160509.png" /></td> </tr></table>

is not identically equal to zero, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605011.png" /> is called a biased estimator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605013.png" /> is called the bias or systematic error of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605014.png" />.

Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605015.png" /> be mutually-independent random variables with the same normal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605016.png" />, and let

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Then the statistic

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is a biased estimator of the variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605019.png" /> since

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that is, the estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605021.png" /> has bias <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605022.png" />. The mean-square error of this biased estimator is

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The best [[Unbiased estimator|unbiased estimator]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605024.png" /> is the statistic

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with mean-square error

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When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605027.png" />, the mean-square error of the biased estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605028.png" /> is less than that of the best unbiased estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605029.png" />.

There are situations when unbiased estimators do not exist. For example, there is no unbiased estimator for the absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605030.png" /> of the mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605031.png" /> of the normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605032.png" />, that is, it is only possible to construct biased estimators for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016050/b01605033.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>