Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means

Details Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means

2009-08-12

arxiv.org/abs/0908.1712v1

Annals of Statistics 2009, Vol. 37, No. 4, 1685-1704

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-AOS630 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/08-AOS630

We consider the classical problem of estimating a vector $\bolds{\mu}=(\mu_1,...,\mu_n)$ based on independent observations $Y_i\sim N(\mu_i,1)$, $i=1,...,n$. Suppose $\mu_i$, $i=1,...,n$ are independent realizations from a completely unknown $G$. We suggest an easily computed estimator $\hat{\bolds{\mu}}$, such that the ratio of its risk $E(\hat{\bolds{\mu}}-\bolds{\mu})^2$ with that of the Bayes procedure approaches 1. A related compound decision result is also obtained. Our asymptotics is of a triangular array; that is, we allow the distribution $G$ to depend on $n$. Thus, our theoretical asymptotic results are also meaningful in situations where the vector $\bolds{\mu}$ is sparse and the proportion of zero coordinates approaches 1. We demonstrate the performance of our estimator in simulations, emphasizing sparse setups. In ``moderately-sparse'' situations, our procedure performs very well compared to known procedures tailored for sparse setups. It also adapts well to nonsparse situations.

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