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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.