= Finite-sample bias in a negative multinomial probability estimator
For $n$ independent one-stop <negative multinomial distribution> observations, the natural estimates are $\widehat\alpha=S_X/(n+S_X+S_Y)$ and $\widehat\beta=S_Y/(n+S_X+S_Y)$. Their sum is $g((S_X+S_Y)/n)$ with strictly concave $g(z)=z/(1+z)$, so <Jensen's inequality> makes its expectation strictly below $\alpha+\beta$. Conditional category proportions split this bias between both estimates. With zero observed counts of a category, that estimate lies at zero; in a parameter space requiring strict positivity the likelihood supremum is not attained.
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