= Gaussian net test
{c}
A <Gaussian net test> turns a finite covering of an alternative into a test with an explicit covering-number cost. In the <Gaussian sequence model> $Y=\theta+n^{-1/2}g$, compare a null $\theta_0$ with centers $\theta_1,\ldots,\theta_N$ in <L2 space> satisfying $\|\theta_j-\theta_0\|_2\geq r>0$. Put $\delta_j=\theta_j-\theta_0$ and reject if any
$$
\langle Y-\theta_0,\delta_j\rangle-\frac12\|\delta_j\|_2^2\geq0.
$$
Each pairing is well-defined as an <isonormal Gaussian process> pairing even though the noise sequence itself is not in <L2 space>. Under the null, it is centered normal with <variance> $\|\delta_j\|_2^2/n$ before subtracting the threshold. The <Gaussian tail bound> and <union bound> give
$$
\mathbb P_{\theta_0}(\text{reject})\leq N e^{-nr^2/8}.
$$
If $\|\theta-\theta_j\|_2\leq r/4$ for some center, the <mean> of that center's displayed statistic is at least $\|\delta_j\|_2^2/4$, by the <Cauchy-Schwarz inequality>. Thus
$$
\mathbb P_\theta(\text{accept})\leq e^{-nr^2/32}.
$$
A radius-$r/4$ net with centers in the separated alternative supplies such a test for its entire covered set. To make the first error small, the squared separation $nr^2$ must dominate $\log N$. This is why metric complexity, rather than just individual Gaussian tails, enters composite testing.
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