Multidimensional two-component Gaussian mixtures detection

Abstract: 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 ($f=\phi_d$) against $f$ is a two-component mixture: $f=(1-\varepsilon)\phi_d +\varepsilon \phi_d (.-\mu)$ where $(\varepsilon,\mu)$ are unknown parameters. Optimal separation conditions on $\varepsilon, \mu, n$ 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.