Estimators and Randomized Procedures — Examples

For each $x\in \mathcal{X}$, the set function $C\mapsto K(x,C)$ is the Dirac measure at $T(x)$, so it is a probability measure by [F2] and [F3].

Fix $C\in \mathcal{B}$ and put $E=T^{-1}(C)\in \mathcal{A}$ by [F1] and [F5]. The evaluation $x\mapsto K(x,C)$ is $1$ on $E$ and $0$ on $\mathcal{X}\setminus E$, which belongs to $\mathcal{A}$ by [F6]. For any Borel subset $D\subseteq [0,1]$, its preimage is one of $\mathrm{\varnothing }$, $\mathcal{X}$, $E$, or $\mathcal{X}\setminus E$, hence is measurable. Thus this evaluation is measurable.

Steps 1.1 and 1.2 verify both clauses of [F4], so $K$ is a probability kernel and therefore a randomized procedure. Since each section is the point mass at $T(x)$, it takes the action $T(x)$ with probability one.

The full power set on the one-point data space is a sigma-algebra by [F7], and $P_{{\theta }_0}={\delta }_{x_0}$ is a probability measure by [F1]. Hence the stated family is a statistical experiment by [F2].

The full power set on $\{0,1\}$ is a sigma-algebra by [F7]. The action section assigns masses $0$, $1-p$, $p$, and $1$ to $\mathrm{\varnothing }$, $\{0\}$, $\{1\}$, and $\{0,1\}$, respectively. They are nonnegative and have total mass one by [A1]. For any disjoint sequence of subsets of $\{0,1\}$, each of the two points occurs in at most one member, so the mass of the union equals the sum of the assigned masses. Thus $K(x_0,\cdot )$ is a probability measure.

The data space has one point, so for every action event $B$ the evaluation $x\mapsto K(x,B)$ is constant; its preimages are empty or the whole data space, both measurable. Step 1.2 and this measurability verify that $K$ is a probability kernel and hence a randomized procedure.

By [F4], the test function is $\phi (x_0)=K(x_0,\{1\})=p$. The action-law identity [F5] gives $Q_{{\theta }_0}(\{1\})=\int K(x,\{1\})P_{{\theta }_0}(dx)$. The integrand is the simple function $p{1}_{\mathcal{X}}$, whose integral is $p{P}_{{\theta }_0}(\mathcal{X})=p$ by [F1] and [F6]. This includes both endpoints $p=0$ and $p=1$.

The family $\mathcal{A}=\{\mathrm{\varnothing },\mathcal{X}\}$ contains $\mathcal{X}$, and its only members have complements in $\mathcal{A}$; any countable union of its members is either empty or $\mathcal{X}$. Hence $\mathcal{A}$ is a sigma-algebra and $(\mathcal{X},\mathcal{A})$ is a measurable space.

In the full action sigma-algebra, $\{1\}$ is measurable, but ${\phi }^{-1}(\{1\})=\{1\}\mathrm{\notin }\mathcal{A}$. Therefore $\phi$ is not measurable by [F5].

The point mass $P_{{\theta }_0}={\delta }_0$ is a probability measure on this measurable space by [F3]. Thus the stated one-parameter family is a statistical experiment.

In this valid experiment, the map is not a statistic by [F6] and is not a randomized test function by [F7], since step 1.2 shows it is not measurable.

If $\phi$ were the action-1 evaluation of a procedure kernel, then $x\mapsto K(x,\{1\})=\phi (x)$ would have to be measurable by [F8], contradicting step 1.2. Thus the map is not a randomized procedure either, proving the stated claim false.