Abstract
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 to finite mixtures models, here the distribution of the hidden variable is not restricted to a finite set but is spread out over a given interval.We propose an approach to construct an orthogonal series estimator of the mixing density f involving Legendre polynomials. 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 the estimator is adaptive over a collection of specific smoothness classes, more precisely, there exists a constant A > 0 such that, when the order m of the projection estimator verifies m ~ A log(n), the estimator achieves the minimax rate over this collection. Other cases are investigated such as Gamma shape mixtures and scale mixtures of compactly supported densities including Betamixtures. Finally, a consistent estimator of the support of the mixing density f is provided.
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Rebafka, T., Roueff, F. Nonparametric estimation of the mixing density using polynomials. Math. Meth. Stat. 24, 200–224 (2015). https://doi.org/10.3103/S1066530715030023
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DOI: https://doi.org/10.3103/S1066530715030023
Keywords
- mixing density
- nonparametric estimation
- exponential mixture
- scale mixture
- Gamma shape mixture
- polynomial approximation
- support estimation