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A concept indicating that the estimator is unbiased in the limit (cf. [[Unbiased estimator|Unbiased estimator]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138201.png" /> be a sequence of random variables on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138202.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138203.png" /> is one of the probability measures in a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138204.png" />. Let a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138205.png" /> be given on the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138206.png" />, and let there be a sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138207.png" />-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a0138209.png" /> the mathematical expectations of which, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382010.png" />, are given. Then, if, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382011.png" />,
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<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/a/a013/a013820/a01382012.png" /></td> </tr></table>
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one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382013.png" /> is a function which is asymptotically unbiased for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382014.png" />. If one calls <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382015.png" /> "observations" and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382016.png" /> "estimators" , one obtains the definition of an asymptotically-unbiased estimator. In the simplest case of unlimited repeated sampling from a population, the distribution of which depends on a one-dimensional parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382017.png" />, an asymptotically-unbiased estimator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382018.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382019.png" />, constructed with respect to the sample size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382020.png" />, satisfies the condition
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A concept indicating that the estimator is unbiased in the limit (cf. [[Unbiased estimator|Unbiased estimator]]). Let  $  X _ {1} , X _ {2} \dots $
 +
be a sequence of random variables on a probability space  $  ( \Omega , S, P ) $,
 +
where  $  P $
 +
is one of the probability measures in a family $ {\mathcal P} $.  
 +
Let a function  $  g(P) $
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be given on the family  $ {\mathcal P} $,  
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and let there be a sequence of $  S $-
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measurable functions  $  T _ {n} ( X _ {1} \dots X _ {n} ) $,
 +
$  n = 1, 2 \dots $
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the mathematical expectations of which,  $  {\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} ) $,
 +
are given. Then, if, as  $  n \rightarrow \infty $,
  
<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/a/a013/a013820/a01382021.png" /></td> </tr></table>
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$$
 +
{\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} )  \rightarrow \
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g (P),\  P \in {\mathcal P} ,
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$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382022.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013820/a01382023.png" />.
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one says that  $  T _ {n} $
 +
is a function which is asymptotically unbiased for the function  $  g $.
 +
If one calls  $  X _ {1} , X _ {2} \dots $"
 +
observations" and  $  T _ {n} $"
 +
estimators" , one obtains the definition of an asymptotically-unbiased estimator. In the simplest case of unlimited repeated sampling from a population, the distribution of which depends on a one-dimensional parameter  $  \theta \in \Theta $,
 +
an asymptotically-unbiased estimator  $  T _ {n} $
 +
for  $  g ( \theta ) $,
 +
constructed with respect to the sample size  $  n $,
 +
satisfies the condition
 +
 
 +
$$
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{\mathsf E} _  \theta  T _ {n} ( X _ {1} \dots X _ {n} )
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\rightarrow  g ( \theta )
 +
$$
 +
 
 +
for any  $  \theta \in \Theta $,  
 +
as $  n \rightarrow \infty $.

Latest revision as of 18:48, 5 April 2020


A concept indicating that the estimator is unbiased in the limit (cf. Unbiased estimator). Let $ X _ {1} , X _ {2} \dots $ be a sequence of random variables on a probability space $ ( \Omega , S, P ) $, where $ P $ is one of the probability measures in a family $ {\mathcal P} $. Let a function $ g(P) $ be given on the family $ {\mathcal P} $, and let there be a sequence of $ S $- measurable functions $ T _ {n} ( X _ {1} \dots X _ {n} ) $, $ n = 1, 2 \dots $ the mathematical expectations of which, $ {\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} ) $, are given. Then, if, as $ n \rightarrow \infty $,

$$ {\mathsf E} _ {P} T _ {n} ( X _ {1} \dots X _ {n} ) \rightarrow \ g (P),\ P \in {\mathcal P} , $$

one says that $ T _ {n} $ is a function which is asymptotically unbiased for the function $ g $. If one calls $ X _ {1} , X _ {2} \dots $" observations" and $ T _ {n} $" estimators" , one obtains the definition of an asymptotically-unbiased estimator. In the simplest case of unlimited repeated sampling from a population, the distribution of which depends on a one-dimensional parameter $ \theta \in \Theta $, an asymptotically-unbiased estimator $ T _ {n} $ for $ g ( \theta ) $, constructed with respect to the sample size $ n $, satisfies the condition

$$ {\mathsf E} _ \theta T _ {n} ( X _ {1} \dots X _ {n} ) \rightarrow g ( \theta ) $$

for any $ \theta \in \Theta $, as $ n \rightarrow \infty $.

How to Cite This Entry:
Asymptotically-unbiased estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotically-unbiased_estimator&oldid=12532
This article was adapted from an original article by O.V. Shalaevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article