Nonparametric estimation of the mixing density using polynomials

We consider the problem of estimating the mixing density $f$ from $n$ i.i.d. observations distributed according to a mixture density with unknown mixing distribution. In contrast with finite mixtures models, here the distribution of the hidden variable is not bounded to a finite set but is spread out over a given interval. An orthogonal series estimator of the mixing density $f$ is obtained as follows. An orthonormal sequence $(\psi_k)_k$ is constructed using Legendre polynomials. Then a standard projection estimator is defined, that is, the first $m$ coefficients of $f$ in this basis are unbiasedly estimated from the observations. The construction of the orthonormal sequence varies from one mixture model to another. Minimax upper and lower bounds of the mean integrated squared error are provided which apply in various contexts. In the specific case of exponential mixtures, it is shown that there exists a constant $A>0$ such that, for $m\sim A \log(n)$, the orthogonal series estimator achieves the minimax rate in a collection of specific smoothness classes, hence, is adaptive over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Beta mixtures.