Unbiased Estimators

Definition

Consider a random variable $x: \Omega \to \R ^n$. The error of the estimate $\xi \in \R ^n$ is the random variable $e: \Omega \to \R ^n$ which is defined by $e(\omega ) = x(\omega ) - \xi $. The bias of an estimate is the expected value of the error. An estimate is unbiased if it has zero bias.

Likewise, if we have another random variable $y: \Omega \to \R ^m$, then the error of the estimator $f: \R ^m \to \R ^n$ is the random variable $e: \Omega \to \R ^n$ defined by $e(\omega ) = f(x(\omega )) - y(\omega )$.