with $Z_1,\ldots,Z_{k-1}$ in a pooled test statistic $Z_{(k)}$, the possibility of early stopping breaks the joint canonical assumption and makes $Z_{(k)}$ non-normally distributed. This non-normality is illustrated with the example of Pocock’s two-stage design in Section \[Pocock\_example\]. Now a few *finite sample* results are presented; asymptotic results are reported in Section \[LSP\].

Conditional and Unconditional Maximum Likelihood Estimation {#Section2.3}
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In GSDs, $\alpha$-spending functions determine cutoff values $\{c_1,\ldots,c_K\}$ that drive decisions to stop at stage $k$ (for $1\le k < K-1$) or continue through stage $K$. These decisions are defined by the events $\left\{\cap_{j=1}^{k-1}\left\{Z_{(j)}\le c_j\right\}\right\}\cap \left\{Z_{(k)}>c_k\right\}$ and $\cap_{j=1}^{K-1}\left\{Z_{(j)}\le c_k\right\}$, respectively, and they are conveniently summarized by a random variable denoting the stopping stage: $$D := K \cdot I\left(\cap_{j=1}^{K-1}\left\{Z_{(j)}\le c_j\right\}\right) 
+ \sum_{k=1}^{K-1} k \cdot I\left(\left(\cap_{j=1}^{k-1}\left\{Z_{(j)}\le c_j\right\}\right)\cap \left\{Z_{(k)}>c_k\right\}\right)$$ defined on $1,\ldots,K$.

The statistic $Z_{(k)}$ is a function of the observations $\textbf{\textit{X}}_{(k)} = (X_1,\ldots,X_{n_{(k)}})$ that were observed prior to stopping. $D$ will appear as random index, such as in $\mathbf{X}_{(D)}$, to underline the fact that the stopping stage is unknown and is described probabilistically though the random variable $D$. The fixed index $k$ in $\mathbf{X}_{(k)}$ indicates that the random variable $D$ took value $k$ (the experiment stopped at stage $k$). Thus, $\mathbf{X}_{(k)}$ is 