 Finite-sample bias in a negative multinomial probability estimator

For $n$ independent one-stop negative multinomial distribution observations, the natural estimates are $\hat{\alpha }=S_X/(n+S_X+S_Y)$ and $\hat{\beta }=S_Y/(n+S_X+S_Y)$. 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.