 Exponential sampling stopped by a data-dependent coin

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 ${\theta }^N{e}^{-\theta S}$. 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 $\theta$.