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Let the regularized incomplete beta and gamma functions be defined as usual: \begin{equation} I_p(z,w) = \frac a {B(z,w)} \int_0^p t^{z-1} (1-t)^{w-1} \,\mathrm dt, \end{equation} \begin{equation} \gamma(r,t) = \frac 1 {\Gamma(r)} \int_0^t s^{r-1} e^{-s} \,\mathrm ds. \end{equation}

Let $r \in \mathbb N$, and consider $a, b > 0$. As is well known, $I_p(r,a/p-b) \rightarrow \gamma(r,a)$ as $p \rightarrow 0$.

In the context of a statistical estimation problem (I can provide details if you are interested), I have been able to prove that the difference between the incomplete beta function and its limit, \begin{equation} I_p(r,a/p-b) - \gamma(r,a), \end{equation} is positive for all $p \in (0,1)$ if $a$ is sufficiently larger than $r$, and negative if $a$ is sufficiently smaller than $r$. For example, if you have access to Matlab you can check that (note that Matlab's gammainc uses reverse order for its arguments)

p = 1e-3;  r = 5; a = 10; b = 5; betainc(p,r,a/p-b) - gammainc(a,r)

is positive, whereas

p = 1e-3;  r = 5; a = 2; b = 5; betainc(p,r,a/p-b) - gammainc(a,r)

is negative.

I'd like to know the widest range of values of $a$ for which that difference is assured to be positive/negative.

So, my question is: Is there a known theorem that gives sufficient conditions on $a, b, r$ that assure that $I_p(r,a/p-b) - \gamma(r,a)$ is positive/negative? Even if you don't know any such theorem, a pointer in the right direction would be appreciated.

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