= Parity estimator for a zero-truncated Poisson count
For the <zero-truncated Poisson distribution>, $2\mathbf1_{\{Y\text{ even}\}}$ is the unique integrable nonrandomized <unbiased estimator> of $p=1-e^{-\theta}$. Indeed unbiasedness gives $\sum_{y\ge1}t(y)\theta^y/y!=e^\theta+e^{-\theta}-2$, and <power series> coefficients force the parity values. Its <variance> is $2p-p^2$; clipping it to $\mathbf1_{\{Y\text{ even}\}}$ gives <mean squared error> $p/2$, strictly smaller for $0<p<1$.
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