Multidimensional two-component Gaussian mixtures detection

Let $(X\mathrm{\_}1,\dots ,X\mathrm{\_}n)$ be a $d$-dimensional i.i.d sample from a distribution with density $f$. The problem of detection of a two-component mixture is considered. Our aim is to decide whether $f$ is the density of a standard Gaussian random $d$-vector ($f=\varphi \mathrm{\_}d$) against $f$ is a two-component mixture: $f=(1-\epsilon )\varphi \mathrm{\_}d+\epsilon \varphi \mathrm{\_}d(\mathrm{.}-\mu )$ where $(\epsilon ,\mu )$ are unknown parameters. Optimal separation conditions on $\epsilon ,\mu ,n$ and the dimension $d$ are established, allowing to separate both hypotheses with prescribed errors. Several testing procedures are proposed and two alternative subsets are considered.