Estimators and Randomized Procedures

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be a statistical experiment (Statistical experiments) and let $(\mathsf{D},\mathcal{G})$ be a measurable action space: a set of available actions equipped with a sigma-algebra of action events, hence a measurable space (Measurable spaces and measurable sets). A randomized procedure (or randomized decision rule) is a probability kernel $K$ from $(\mathcal{X},\mathcal{A})$ to $(\mathsf{D},\mathcal{G})$, in the pointwise sense of Measure kernel and probability kernel. Thus $K(x,\cdot )$ is a probability measure on $(\mathsf{D},\mathcal{G})$ for every $x\in \mathcal{X}$, and $x\mapsto K(x,B)$ is $\mathcal{A}$-measurable for every $B\in \mathcal{G}$. After observing $x$, the procedure assigns probability $K(x,B)$ to taking an action in $B$. The kernel is specified without reference to the unknown parameter. This definition imposes no measurable structure on $\mathrm{\Theta }$. The kernel is the specification of the procedure; no representation by a separate random seed is asserted for arbitrary measurable action spaces.

Let $K$ be a randomized procedure for the experiment $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ with action space $(\mathsf{D},\mathcal{G})$. Give $[0,+\mathrm{\infty }]$ its Borel sigma-algebra. For every nonnegative $\mathcal{G}\mathrm{/}\mathcal{B}([0,+\mathrm{\infty }])$- measurable function $g:\mathsf{D}\to [0,+\mathrm{\infty }]$, the function $h_g(x)={\int }_{\mathsf{D}}g(a)K(x,da)$ is $\mathcal{A}$-measurable. In particular, $x\mapsto K(x,B)$ is measurable for every action event $B\in \mathcal{G}$. For each $\theta \in \mathrm{\Theta }$, the set function $Q_{\theta }(B)={\int }_{\mathcal{X}}K(x,B)P_{\theta }(dx),B\in \mathcal{G},$ is a probability measure on $(\mathsf{D},\mathcal{G})$, and for every such $g$, ${\int }_{\mathsf{D}}gd{Q}_{\theta }={\int }_{\mathcal{X}}({\int }_{\mathsf{D}}g(a)K(x,da))P_{\theta }(dx)\mathrm{.}$ with nonnegative extended values allowed.

Let $\mathsf{E}=((\mathcal{X},\mathcal{A}),\mathrm{\Theta },(P_{\theta }{)}_{\theta \in \mathrm{\Theta }})$ be an experiment (Statistical experiments). A statistic with target measurable space $(\mathcal{Y},\mathcal{B})$ is a measurable map $T:(\mathcal{X},\mathcal{A})\to (\mathcal{Y},\mathcal{B})$, where measurability means that $T^{-1}(B)\in \mathcal{A}$ for every $B\in \mathcal{B}$ (A measurable function between measurable spaces). If $\psi :\mathrm{\Theta }\to \mathcal{Y}$ is an estimand (Estimands on a statistical model) and its target set $\mathcal{Y}$ is equipped with $\mathcal{B}$, a statistic $\widehat{\psi }$ with this same target space is an estimator of $\psi$. The value $\widehat{\psi }(x)$ is the estimate at observed data $x$. Giving $\mathcal{Y}$ a measurable structure is an additional choice; this definition does not assert that $\psi$ is measurable as a function of the parameter.

A randomized test is a randomized procedure with binary action space whose action set is $\{0,1\}$, equipped with its full sigma-algebra $2^{\{0,1\}}$ (the power set, which is a sigma-algebra by Sigma-algebras), where $1$ means reject and $0$ means do not reject (Randomized procedures as probability kernels). Its test function is $\phi (x)=K(x,\{1\}),x\in \mathcal{X}\mathrm{.}$ Since $K(x,\cdot )$ is a probability measure and $\{0\},\{1\}$ partition the action set, $K(x,\{0\})+K(x,\{1\})=1$. Thus $\phi (x)\in [0,1]$ and $1-\phi (x)=K(x,\{0\})$ is the probability of not rejecting. Give $[0,1]$ the trace of the Borel sigma-algebra on $[0,+\mathrm{\infty }]$, namely $\{C\cap [0,1]:C\in \mathcal{B}([0,+\mathrm{\infty }])\}$. By A randomized procedure has a measurable action law, $x\mapsto K(x,\{1\})$ is measurable; since it has range in $[0,1]$, its preimage of $C\cap [0,1]$ equals its preimage of $C$ and is measurable for every such $C$. Hence $\phi$ is a measurable map in the preimage sense of A measurable function between measurable spaces. Its value is the rejection probability after observing $x$. The convention $1=\text{reject}$ is fixed here. No null or alternative hypothesis is part of this definition.