Does Hoeffding or Bernstein inequality provide a more accurate bound for this problem?

Let $$X_i$$, $$i=1,\dots, n$$ be independent random variables with $$EX_i = \mu_i$$, $$\operatorname{Var}X_i=\sigma^2$$ and $$|X_i - \mu_i| \leq b$$, $$i=1,\dots,n$$. Let $$X=\sum X_i$$ and $$\mu = \sum \mu_i$$. Let $$\tau>0$$ be a fixed constant. Find $$t$$ such that $$P(|X-\mu|\leq t) \geq 1 - 2n^{-\tau}$$ using the Hoeffding and the Bernstein inequalities. Which of the inequalities provides a more accurate bound?

The Hoeffding inequality states that $$P(\bar{X}-E[\bar{X}] \geq t) \leq e^{-2nt^2}$$ where $$t\geq0$$, and $$\bar{X}$$ denotes the empirical mean of these variables.

Whereas, the Bernstein inequality states that $$P(X-\mu\geq t) \leq \exp\left(\frac{-t^2}{2(\sigma^2+bt)}\right).$$

I am not sure how to proceed with this problem.

• Presumably "more accurate" here means whichever gives the bound with smallest value of $t$. You want $P(|X-\mu|\geq t)\leq 2n^{-\tau}$, so figure out which bound gives you a smaller $t$. Intuitively a smaller $t$ implies the empirical mean is more concentrated around the true mean. – Alex R. Oct 15 '18 at 23:43