Adaptive density estimation using finite Gaussian mixtures

The maximum likelihood estimator proposed in Maugis and Michel (2009) is here considered for the estimation of univariate densities. This estimator is a finite Gaussian mixture whose number of components is selected using a non asymptotic penalized criterion and it fulfills an oracle inequality. In this work, considering a collection of univariate densities whose logarithm is locally $\beta$-H\"older with moment and tail conditions, we show that the penalized estimator is adaptive to the $\beta$ regularity of such densities for the minimax Hellinger risk.