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: $f=(1-\varepsilon)\phi_{d}+\varepsilon\phi_{d}(\cdot -\mu)$ 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.