a random sample $\textbf{X}=\left(X_1,\ldots,X_n\right)$ is $$\begin{aligned}
{\cal{L}}\left(\theta|\textbf{\textit{x}}\right) = \prod_{i=1}^n f_X\left(\textbf{\textit{x}}\right) &= 
h\left(\textbf{\textit{x}}\right) g \left(T\left(\textbf{\textit{x}}\right) |\theta \right) = 
h\left(\textbf{\textit{x}}\right)  e^{\eta(\theta)T\left(\textbf{\textit{x}}\right) + A\left(\theta\right)}\end{aligned}$$ where all relevant information about $\theta$ is absorbed by a sufficient statistic $T\left(\textbf{\textit{x}}\right)$. Assume the test statistic $Z$ is a one-to-one transformation of $T$.

If $LR(t):=g(t|\theta)/g(t|\theta_0)$ is a monotone likelihood ratio (MLR) in $t$, then the Karlin-Rubin theorem allows the construction of uniformly most powerful tests. Using the notation in Section \[normality\], $$\begin{aligned}
{\cal{L}}^c\left(\theta|k,\textbf{\textit{x}}_{(k)}\right) 
&:=& \frac{I\left(D=k\right)}{\text{Pr}_{\theta}\left(D=k\right)}h\left(\textbf{\textit{x}}_{(k)}\right) 
e^{\eta(\theta)T\left(\textbf{\textit{x}}_{(k)}\right) + A\left(\theta\right)} \notag \\
&=& I\left(D=k\right) h\left(\textbf{\textit{x}}_{(k)}\right) 
e^{\eta(\theta)T\left(\textbf{\textit{x}}_{(k)}\right) + A\left(\theta\right) + \log \text{Pr}_{\theta}\left(D=k\right)}, \label{Lc}\\
{\cal{L}}\left(\theta|k,\textbf{\textit{x}}_{(k)}\right) 
&:=& I\left(D=k\right) h\left(\textbf{\textit{x}}_{(k)}\right) e^{\eta(\theta)T\left(\textbf{\textit{x}}_{(k)}\right) + A\left(\theta\right)},
\label{Lu}\end{aligned}$$ with associated likelihood ratios: $$\begin{aligned}
LR^c(t) := \frac{{\cal{L}}^c\left(\theta|k,\textbf{\textit{x}}_{(k)}\right)}{{\cal{L}}^c\left(\theta_0|k,\textbf{\textit{x}}_{(k)}\right)} 
\quad\textrm{ and }\quad
LR(t) := \frac{{\cal{L}}\left(\theta|k,\textbf{\textit{x}}_{(k)}\right)}{{\cal{L}}\left(\theta_0|k,\textbf{\textit{x}}_{(k)}\right)}.
\notag\end{aligned}$$ For every $D=k$, $LR^c(t) = LR(t)= \exp \left[\left(\eta(\theta)-\eta(\theta_0)\right) T_{(k)} + \left(A(\theta)-A(\theta_0)\right)\right]$ which means that *the MLR property is preserved with early stopping*.

\[theorem1\] For any fixed $(\alpha_1,\ldots,\alpha_K)$ and $\left(n_1,\ldots,n_K\right)$ $(1)$ $\{T_{(k)} > c_k\}$ is a UMP $\alpha_k$-level test and $(2)$ no test is more powerful than $\{T_{(D)} > c_D\}$.

**Proof.** $(1)$ As shown above, conditional on $D=k$, $T_{(k)}$ is sufficient in exponential families and the MLR property is preserved. By the Karlin-Rubin theorem, the test based on $T_{(k)}$, is uniformly most powerful.\
$(2)$ Using Table 