 Gaussian net test

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 ${\theta }_1,\dots ,{\theta }_N$ in L2 space satisfying $\Vert {\theta }_j-{\theta }_0{\Vert }_2\ge r>0$. Put ${\delta }_j={\theta }_j-{\theta }_0$ and reject if anyEach 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 $\Vert {\delta }_j{\Vert }_2^2/n$ before subtracting the threshold. The Gaussian tail bound and union bound giveIf $\Vert \theta -{\theta }_j{\Vert }_2\le r/4$ for some center, the mean of that center's displayed statistic is at least $\Vert {\delta }_j{\Vert }_2^2/4$, by the Cauchy-Schwarz inequality. ThusA 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 $n{r}^2$ must dominate $\log N$. This is why metric complexity, rather than just individual Gaussian tails, enters composite testing.