# Minimax Test for Heterogeneous Gaussian Mean Shifting

For $i=1,…,N$, we have the data $X_i \sim \mathcal{N}( \delta \mu_i, \mu_i^2 )$, where $\mu_i>0$ and they are some unknown nuisance parameters. We want to test if there is a shift of the means, i.e. $\delta = 0$ v.s. $\delta \neq 0$. Note that in this model, although data points are not iid, there is an invariance: $Pr(X_i>0)$ are same for all $i$.

We are looking for the minimax test w.r.t. all possible values of $\mu_i$'s, i.e.

\begin{align} &\sup_{\phi} \inf_{\mu_1,…,\mu_N} \mathbb{E}_1[\phi(X_1,...,X_N)]\\ &s.t. \sup_{\mu_1,…,\mu_N} \mathbb{E}_0[\phi(X_1,...,X_N)] \leq \alpha, \end{align} where $\phi$ means a test with probablity $\phi(X_1,...,X_N)$ to reject the null.

We believe the solution is the sign test. The statistic is $Y=\sum_{i=1}^N 1_{\{X_i> 0\}}$, where $1_{\{X_i> 0\}}$ is the indicator function, taking value $1$ if $X_i > 0$ and $0$ otherwise. Assume $N$ is large. The $\alpha$-level test rejects when

\begin{equation} \vert 0.5 \sqrt{N} ( Y/N - 0.5 ) \vert > C_{1-\alpha/2}, \end{equation} where $C_{1-\alpha/2}$ is the $(1-\alpha/2)$-th quantile of standard normal. If we let $\delta=\epsilon/\sqrt{N}$, for some constant $\epsilon$, the above test yields an asymptotic power $1 - \Phi( C_{1-\alpha/2} - \epsilon / \sqrt{\pi \gamma} ) + \Phi( -C_{1-\alpha/2} - \epsilon / \sqrt{\pi \gamma} )$. That is, regardless the value of $\mu_i$'s, the $\alpha$-level sign test has a same asymptotic power.

So far we are trying to prove this test is indeed minimax. However, I find the minimax hypothesis testing is a very narrow topic, and hardly did I find any useful reference. I am pretty confused if I am on the right track or if this problem has been solved before. Any suggestions or references would be helpful.