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

*(English)*Zbl 1166.62005Summary: We consider the classical problem of estimating a vector \(\mu=(\mu_1,\dots,\mu_n)\) based on independent observations \(Y_i\sim N(\mu_i, 1)\), \(i=1,\dots,n\). Suppose \(\mu_i\), \(i=1,\dots,n\), are independent realizations from a completely unknown \(G\). We suggest an easily computed estimator \(\widehat\mu\), such that the ratio of its risk \(E(\widehat\mu-\mu)^2\) 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 \(\mu\) 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.

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 \(\mu\) 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.

##### MSC:

62C12 | Empirical decision procedures; empirical Bayes procedures |

62C25 | Compound decision problems in statistical decision theory |

62H12 | Estimation in multivariate analysis |

##### Software:

EBayesThresh
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\textit{L. D. Brown} and \textit{E. Greenshtein}, Ann. Stat. 37, No. 4, 1685--1704 (2009; Zbl 1166.62005)

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