Tail bounds for Beta distribution

Question

Say $X\sim\mathrm{Beta}(\alpha,\beta)$. Are there any "nice" closed form upper bounds for the tail probability $P(X\geq\epsilon)$, that are reasonably tight when $\beta$ is large? By "nice" I mean involving only elementary functions, and not for instance the incomplete beta function.

In my specific setting I have $\alpha=1/2$ and $\beta=d/2$ for some (typically large) integer $d$, and so one way of bounding the tail is to write $X=\frac{Y}{Y+Z}$ where $Y\sim\chi^2_1$ and $Z\sim\chi^2_d$, and using an upper bound on $Y$ and lower bound on $Z$ (e.g. from here: What are the sharpest known tail bounds for $\chi_k^2$ distributed variables?). This gives an answer, but it's somewhat messy (requiring the small $d$ case to be handled separately, etc). Is there a cleaner way?

Thanks.

Answer 1

We write $I_{\epsilon}(A,B,\epsilon)$ the incomplete beta function. Thus $P(X>\epsilon) = 1-I_{\epsilon}(A,B,\epsilon)$

A simple bound is

$$\frac{2\sqrt{\epsilon}}{2\sqrt{\epsilon}+B\left(\frac{1}{2},\frac{d}{2}\right)}< I_\epsilon\left(\frac{1}{2},\frac{d}{2},\epsilon\right)<\frac{2 \sqrt{\epsilon}}{B\left(\frac{1}{2},\frac{d}{2}\right)}$$

A tighter lower bound (so upper bound for the tail) would be

$$\frac{2 \left(\sqrt{\epsilon} B\left(\frac{1}{2},\frac{d}{2}\right)+2 \epsilon\right)}{2 \sqrt{\epsilon} B\left(\frac{1}{2},\frac{d}{2}\right)+B\left(\frac{1}{2},\frac{d}{2}\right)^2+4 \epsilon}$$

Mathematica will spit them out for you with

1 - 1 / Normal[Series[1/(1 - BetaRegularized[\epsilon, 1/2, d/2]), {\epsilon, 0, n}]]