Multidimensional two-component Gaussian mixtures detection

Abstract

Let $(X\_1,\dots ,X\_n)$ 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 ($f=\varphi \_d$) against $f$ is a two-component mixture: $f=(1-\epsilon )\varphi \_d+\epsilon \varphi \_d(.-\mu )$ 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.