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 $.