Non-asymptotic detection of two-component mixtures with unknown means

This work is concerned with the detection of a mixture distribution from a $\mathbb{R}$-valued sample. Given a sample $X_1,\dots, X_n$ and an even density $\phi$, our aim is to detect whether the sample distribution is $\phi(.-\mu)$ for some unknown mean $\mu$, or is defined as a two-component mixture based on translations of $\phi$. In a first time, a non-asymptotic testing procedure is proposed and we determine conditions under which the power of the test can be controlled. In a second time, the performances of our testing procedure are investigated in 'benchmark' asymptotic settings. A simulation study provides comparisons with classical procedures.