# Asymptotically-unbiased estimator

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=45236