TAOCP 3.3.2 Exercise 15
In the maximum-of-$t$ test, the $j$th observation is V_j=\max(U_{tj},U_{tj+1},\ldots,U_{tj+t-1}).
Section 3.3.2: Empirical Tests
Exercise 15. [M10] In the maximum-of-$t$ test, why are $V_0', V_1', \ldots, V_{n-1}'$ supposed to be uniformly distributed between zero and one?
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In the maximum-of-$t$ test, the $j$th observation is
$$ V_j=\max(U_{tj},U_{tj+1},\ldots,U_{tj+t-1}). $$
For $0\le v\le1$,
$$ \Pr(V_j\le v) =\Pr(U_{tj}\le v,\ldots,U_{tj+t-1}\le v). $$
Since the $U$'s are independent and uniformly distributed on $[0,1]$,
$$ \Pr(V_j\le v)=v^t. $$
Thus the distribution function of $V_j$ is $F(v)=v^t$.
The test uses
$$ V'_j=V_j^{,t}. $$
Hence, for $0\le x\le1$,
$$ \Pr(V'_j\le x) =\Pr(V_j^{,t}\le x) =\Pr(V_j\le x^{1/t}) =(x^{1/t})^t =x. $$
Therefore the distribution function of $V'_j$ is $x$ on $[0,1]$, which is the uniform distribution. Consequently $V'_0,V'1,\ldots,V'{n-1}$ are supposed to be uniformly distributed between zero and one. ∎