Multidimensional two-component Gaussian mixtures detection

Let $(X\_1,\ldots,X\_n)$ 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 ($f=ϕ\_d$) against $f$ is a two-component mixture: $f=(1-\varepsilon)ϕ\_d +\varepsilon ϕ\_d (.-μ)$ where $(\varepsilon,μ)$ are unknown parameters. Optimal separation conditions on $\varepsilon, μ, n$ 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.

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