Bias and mean squared error

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment: a measurable data space, a parameter-label set, and a family of probability measures on the data space. Let $\psi :\mathrm{\Theta }\to \mathbb{R}$ be its real-valued target function (estimand). An estimator $T:\mathcal{X}\to \mathbb{R}$ is a measurable statistic with the real Borel target (Statistics and estimators). At a parameter $\theta$ for which $T$ is integrable under $P_{\theta }$, its bias is ${\mathrm{Bias}}_{\theta }(T):={\mathbb{E}}_{\theta }[T]-\psi (\theta )\mathrm{.}$

Its mean squared error (MSE) is the extended nonnegative number ${\mathrm{MSE}}_{\theta }(T):={\mathbb{E}}_{\theta }[(T-\psi (\theta ){)}^2]\in [0,+\mathrm{\infty }]\mathrm{.}$ This is the risk of the deterministic rule induced by $T$ under squared-error loss $L(\theta ,a)=(a-\psi (\theta ){)}^2$: its Dirac kernel is constructed in A statistic induces a deterministic probability kernel, and the risk supplier's integral formula reduces to the displayed expectation when the action is $T(x)$ by Integration against a Dirac measure evaluates at its point (Loss functions and risk). For each fixed $\theta$, this loss is Borel in the real action $a$; no measurable structure on $\mathrm{\Theta }$ or measurability of $\psi$ is required. The expectation convention is Expectation of a nonnegative or integrable random variable. If $T$ is square-integrable under $P_{\theta }$, then its variance is defined by Moments, variance, and covariance on a probability space and this MSE is finite. Bias is not defined at parameters where $T$ is not integrable.