Solved – Tail bounds for Beta distribution

Say $Xsimmathrm{Beta}(alpha,beta)$. Are there any "nice" closed form upper bounds for the tail probability $P(Xgeqepsilon)$, 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 $Ysimchi^2_1$ and $Zsimchi^2_d$, 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?


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

$$frac{2sqrt{epsilon}}{2sqrt{epsilon}+Bleft(frac{1}{2},frac{d}{2}right)}< I_epsilonleft(frac{1}{2},frac{d}{2},epsilonright)<frac{2 sqrt{epsilon}}{Bleft(frac{1}{2},frac{d}{2}right)}$$

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

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

Mathematica will spit them out for you with

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

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