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 $n \rightarrow \infty$.