I need help to how to find $a_\alpha(\bar{X}), b_\alpha(\bar{X})$. I don't know how to find infimum and supremum which satisfies in both cases. I would appreciate any help or hint.
Let $\wp$ be a family of probability distribution on $[0,1]$, and for $P \in \wp$ let $\mu(P) := \int xP(dx)$.
Further, let
$$\wp(\mu) := \left \{P \in \wp : \mu(P)=\mu \right \}$$
for any $\mu$ in the set $\left \{ \mu(P) : P \in \wp \right \} \subset [0,1].$
$X_1, X_2, ...$ independent random variables with distribution $P=(1-\mu)\delta_0+\mu \delta_1$.
$$a_\alpha(\bar{X}):=\inf \left \{ \mu \leq \bar{X}: H^{*}(\bar{X},\mu) < -\log(\alpha)/n \right \} $$
$$b_\alpha(\bar{X}):=\sup\left \{ \mu \geq \bar{X}: H^{*}(\bar{X},\mu) < -\log(\alpha)/n \right \} $$ where $H^{*}(r,\mu)=r\log(\frac{r}{\mu})+(1-r)r\log(\frac{1-r}{1-\mu})$