Group-sequential alpha spending — Examples

Since $\exp (0)=1$ by its power series and $\exp$ is strictly increasing by [F9], $e=\exp (1)>1$. Thus $1+(e-1)t$ increases from $1$ to $e$ as $t$ goes from $0$ to $1$. By [F7]–[F8], $A_P(0)=0$, $A_P(1)=\alpha \log e=\alpha$, and $A_P$ is nondecreasing with values in $[0,\alpha ]$. At $t_1=1\mathrm{/}2$ this gives $a_1=\alpha \log ((1+e)\mathrm{/}2)$ and $0<a_1<\alpha$.

The standard normal CDF satisfies $\mathrm{\Phi }(y)-\mathrm{\Phi }(x)={\int }_x^y\varphi (s)ds$ for $x<y$. By [F4]–[F5], this difference is positive and at most $(y-x)\mathrm{/}\sqrt{2\pi }$, so $\mathrm{\Phi }$ is strictly increasing and continuous. Its limits at $-\mathrm{\infty }$ and $+\mathrm{\infty }$ are $0$ and $1$: [F10] applied to $(-n,\mathrm{\infty })$ gives the lower-tail limit, and [F10] applied to $(-\mathrm{\infty },n]$ gives the upper-tail limit; the density has no point masses. Therefore [F11] gives a unique $q(u)$ for every $u\in (0,1)$. In particular $c_1=q(1-a_1\mathrm{/}2)$ is finite, positive, and symmetry and atomlessness give $P_0(\mathrm{\mid }{Z}_1\mathrm{\mid }\ge c_1)=a_1$.

The canonical covariance matrix is $\mathrm{\Sigma }=\left(\begin{array}{cc}1& 1\mathrm{/}\sqrt{2}\\ 1\mathrm{/}\sqrt{2}& 1\end{array}\right)$, so its determinant is $1\mathrm{/}2>0$ and the correlation is $1\mathrm{/}\sqrt{2}$. Applying [F2]–[F3] at the second look gives the unique finite $c_2$ satisfying the displayed cumulative two-sided crossing equation. The event $\{\mathrm{\mid }{Z}_1\mathrm{\mid }\ge c_1\}$ and the continued-path event $\{\mathrm{\mid }{Z}_1\mathrm{\mid }<c_1,\mathrm{\mid }{Z}_2\mathrm{\mid }\ge c_2\}$ are disjoint and their union is the final rejection event. Hence the continued-path event has joint probability $\alpha -a_1$, and since $P_0(\mathrm{\mid }{Z}_1\mathrm{\mid }<c_1)=1-a_1>0$, its conditional probability given continuation is $(\alpha -a_1)\mathrm{/}(1-a_1)$. The total rejection probability is $A_P(1)=\alpha$.

Define $\mathrm{\Phi }(x)={\int }_{-\mathrm{\infty }}^x\varphi (s)ds$. For $x<y$, positivity in [F5] gives $\mathrm{\Phi }(y)-\mathrm{\Phi }(x)>0$, while $\varphi (s)\le 1\mathrm{/}\sqrt{2\pi }$ gives $\mathrm{\Phi }(y)-\mathrm{\Phi }(x)\le (y-x)\mathrm{/}\sqrt{2\pi }$. Thus $\mathrm{\Phi }$ is strictly increasing and continuous.

The density $\varphi$ is even, so $\mathrm{\Phi }(0)=1\mathrm{/}2$. Applying [F6] to the increasing upper half-lines $(-n,\mathrm{\infty })$ gives $\mathrm{\Phi }(x)\to 0$ as $x\to -\mathrm{\infty }$; applying it to the increasing lower half-lines $(-\mathrm{\infty },n]$ gives $\mathrm{\Phi }(x)\to 1$ as $x\to +\mathrm{\infty }$. Hence [F7] gives a unique quantile $z_p$ for each $p\in (0,1)$, and $z_{0.975}>0$.

For $0<t\le 1$, $z_{0.975}\mathrm{/}\sqrt{t}$ decreases as $t$ increases; since $\mathrm{\Phi }$ increases, $A_{OF}(t)$ is nondecreasing. Its values are positive and at most $2[1-\mathrm{\Phi }(z_{0.975})]=0.05$, and its limit at $0$ is $0$ by the upper-tail limit in step 2.1. Thus the declared $A_{OF}(0)=0$ makes it admissible. At $t_1=1\mathrm{/}2$, its value is $a_1=2[1-\mathrm{\Phi }(z_{0.975}\sqrt{2})]$, with $0<a_1<0.05$.

By symmetry of the density in [F4], $P_0(\mathrm{\mid }{Z}_1\mathrm{\mid }\ge c)=2[1-\mathrm{\Phi }(c)]$ for $c\ge 0$. Therefore $c_1=z_{1-a_1\mathrm{/}2}$ has first-look crossing probability $a_1$. The canonical covariance matrix is $\mathrm{\Sigma }=\left(\begin{array}{cc}1& 1\mathrm{/}\sqrt{2}\\ 1\mathrm{/}\sqrt{2}& 1\end{array}\right)$, so its determinant is $1\mathrm{/}2>0$ and the correlation is $1\mathrm{/}\sqrt{2}$. Applying [F2]–[F3] at the second look gives the unique finite $c_2$ satisfying the displayed cumulative two-sided crossing equation, with total probability $A_{OF}(1)=0.05$.