first $n_1$, $n_1+n_2$, and $n_1+n_2+n_3$ observations, respectively. Resolving (\[GSnew\]) numerically, under $X \sim N(\theta,1)$ and $\alpha_0 \approx 0.0172$, one finds $n_1=98$ and $c_1=2.12$ for stage 1, $n_2=98$ and $c_2 = 2.01$ for stage 2, and $n_3=576$ and $c_3 = 2.02$ for stage 3. Using these stage-specific sample sizes and critical values, Table \[table\_gs\] reports operational characteristics of the new GSD based on a Monte-Carlo resampling with $100,000$ iterations.

   $\theta$   $\text{Pr}_{\theta}\left({\cal{R}}_1 \vert \theta \right)$   $\text{Pr}_{\theta}\left({\cal{R}}_2 \vert \theta \right)$   $\text{Pr}_{\theta}\left({\cal{R}}_3 \vert \theta \right)$   $\text{E}[N\vert \theta]$  
  ---------- ------------------------------------------------------------ ------------------------------------------------------------ ------------------------------------------------------------ --------------------------- --
     0.0                                0.0170                                                       0.0336                                                     **0.0509**                                     $751$            
     0.1                                0.1287                                                       0.3044                                                     **0.7983**                                     $584$            
     0.2                                0.4424                                                     **0.8016**                                                     0.9998                                       $267$            
     0.3                              **0.8018**                                                     0.9877                                                       1.0000                                       $125$            

  : \[table\_gs\] Stage-specific probabilities of rejecting $H_0$. $\text{E}[N\vert \theta]$ is the average sample size.

As it will be shown in Section \[mostpower\], if each $Z_{(k)}$ is a continuous function of a monotone likelihood ratio, then the proposed test is the most powerful against any ordered alternatives $\theta_1 > \theta_2 > \cdots > \theta_K> 0$ given an $\alpha$-spending function.

Most Powerful Group Sequential Tests for Ordered Alternatives {#mostpower}
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Let $X\sim f_X(\theta)$, where $f$ belongs to the exponential family. Without a possibility of early stopping, the likelihood for a realization $\textbf{\textit{x}}=\left(x_1,\ldots,x_n\right)$ of 