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A deconvolution path for mixtures. (English) Zbl 1404.62033
In this paper the authors propose a nonparametric method for deconvolution that is both statistically and computationally efficient. The method is motivated in terms of an underlying Bayesian model incorporating a prior into the model \[ y_i|\mu_i\sim\phi(y_i|\mu_i),\quad \mu_i\sim f_0,\;\;\mu_i\;\;\text{i.i.d.} \]
Instead using a full Bayes analysis the authors use a two-step “bin and smooth” procedure, which in the “bin” step forms a histogram of the sample, yielding the number of observations \(x_j\) that fall into the \(j\)-th histogram bin. In the “smooth” step these counts are used to compute a maximum a posteriori (MAP) estimate of \(f_0\) under a prior that encourages smoothness.
It is shown that the proposed nonparametric empirical-Bayes procedure yields excellent performance for deconvolution, at reduced computational cost compared to full nonparametric Bayesian methods. The main theorem establishes conditions under which the method yields a consistent estimate of the mixing distribution \(f_0\). Simulation evidence that the method offers practical improvements over existing state-of-the-art methods is also provided.

MSC:
62G07 Density estimation
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62G20 Asymptotic properties of nonparametric inference
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