Squared-error risk for a Bernoulli observation

For each $\theta \in [0,1]$, define $P_{\theta }(A)=\theta 1_{\{1\in A\}}+(1-\theta )1_{\{0\in A\}}$ for $A\subseteq \{0,1\}$. Its singleton masses are nonnegative and sum to one; any disjoint sequence of subsets has at most two nonempty terms, and their masses add to the union mass. Thus $P_{\theta }$ is a probability measure by [F1]. The identity rule $d(x)=x$ is measurable on the finite full-power-set spaces and estimates the target $\psi (\theta )=\theta$ by [F2].

The finite expectation formula [F4] and risk definition [F3] give $R(\theta ,d)=(1-\theta )(0-\theta {)}^2+\theta (1-\theta {)}^2=\theta (1-\theta )$.

Since $\theta (1-\theta )=\frac{1}{4}-(\theta -\frac{1}{2}{)}^2$, the risk is at most $1\mathrm{/}4$, with equality at $\theta =1\mathrm{/}2$; substitution gives risk zero at $\theta =0$ and $\theta =1$.