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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.

Write $r=1-\alpha -\beta >0$. Differentiating the probability generating function at $(1,1)$ givesExpanding its denominator as a geometric series and then applying the binomial theorem givesCoefficient identification proves $\boxed{\mathrm{Pr}(X=x,Y=y)=r\left(\genfrac{}{}{0ex}{}{x+y}{x}\right){\alpha }^x{\beta }^y}$, a one-stop negative multinomial distribution.

For an independent identically distributed sample let $S_X=\sum X_i$ and $S_Y=\sum Y_i$. Ignoring parameter-independent factors, the log-likelihood is $\ell =n\ln r+S_X\ln \alpha +S_Y\ln \beta$. When both sums are positive, its stationary equations are $S_X/\alpha =n/r$ and $S_Y/\beta =n/r$. Together with $r+\alpha +\beta =1$, these giveThe Hessian is $-(n/r^2)11^T-\mathrm{diag}(S_X/{\alpha }^2,S_Y/{\beta }^2)$, negative definite in this case, so the stationary point is the unique global maximum.

There is an important boundary qualification: if $S_X=0$, the derivative with respect to $\alpha$ is $-n/r<0$, so the likelihood increases as $\alpha$ decreases to zero. The displayed formula then gives a boundary estimator, not an attained MLE in the stated open parameter space $\alpha ,\beta >0$. The analogous issue holds for $S_Y=0$. The formulas are MLEs on the extension allowing zero category probabilities, and interior MLEs when both counts are positive.

For bias, put $p=\alpha +\beta$ and $U=(S_X+S_Y)/n$. Then $\mathbb{E}[U]=p/(1-p)$ and $\hat{\alpha }+\hat{\beta }=g(U)$ with $g(u)=u/(1+u)$. Its second derivative is $-2/(1+u{)}^3<0$, and $U$ is nondegenerate. Strict Jensen's inequality givesIn fact both estimates are downward biased. Conditional on $X_i+Y_i$, the first count is binomial with success probability $\alpha /p$; hence conditional on $S=S_X+S_Y$, $S_X$ is also binomial with that probability. ThereforeThis proves the finite-sample bias in a negative multinomial probability estimator, using the boundary-extended formulas whenever necessary.

Finally, the strong law of large numbers applies because the nonnegative variables have finite means: almost surely $\overline{X}\to \alpha /r$ and $\overline{Y}\to \beta /r$. Continuity of the estimator function then givesand similarly for $\hat{\beta }$. Thus the estimates are strongly consistent despite finite-sample bias. Almost surely both sample sums eventually become positive, so the open-parameter MLE issue affects finite samples rather than this asymptotic conclusion.