= Exponential sampling stopped by a data-dependent coin
{title2=$L(\theta;\mathbf x)\propto\theta^Ne^{-\theta\sum X_i}$}
For independent exponential observations of rate $\theta$, stopping after observation $x$ with known probability $\alpha(x)\in(0,1)$ gives a complete-data likelihood proportional to $\theta^Ne^{-\theta S}$. Its minimal sufficient statistic is $(N,S)$ by the parameter-free likelihood-ratio criterion. If only the last value $Y$ is recorded, its density instead is $\alpha(y)e^{-\theta y}/\int_0^\infty\alpha(x)e^{-\theta x}\,dx$. The stopping rule therefore changes the coarsened-data inference even though it adds no complete-data likelihood factor involving $\theta$.
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