NONPARAMETRIC EMPIRICAL BAYES AND COMPOUND DECISION APPROACHES TO ESTIMATION OF A HIGH-DIMENSIONAL VECTOR OF NORMAL MEANS

By Lawrence D. Brown, Eitan Greenshtein. 2009

164 citations recorded by the source corpus.

We consider the classical problem of estimating a vector μ = (μ 1 ,..., μ n ) based on independent observations Y i ∼ N(μ i , 1), i = 1,..., n. Suppose μ i , i = 1,..., n are independent realizations from a completely unknown G. We suggest an easily computed estimator μ, such that the ratio of its risk E(μ ― μ) 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 μ 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.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/25f7719223bcb54c0dafa478f6d73307e9fed007

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. van der Vaart
2. Ryan Martin
3. Stephen G. Walker
4. I. Castillo
5. Eitan Greenshtein
6. Yaacov Ritov
7. Cun-Hui Zhang
8. Roger W. Koenker

In Statistics and Probability

1. A. van der Vaart
2. I. Castillo
3. Ryan Martin
4. Stephen G. Walker
5. Jiaying Gu
6. Cun-Hui Zhang
7. Victor Chernozhukov
8. Richard Nickl

Citations in the corpus

• Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences (2012, background): “Other related papers include [19], [6], [7], [14], [15], [5].”
• Forecasting With Dynamic Panel Data Models (2017, unknown): “Our main theorem extends a result from Brown and Greenshtein (2009) for a vector of means to a panel data model with estimated common coefficients. Importantly, this result also covers the case in which the distribution π(λi|·) degenerates to a point mass. As in Brown and Greenshtein (2009), we are ”
• Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector (2013, methodology): “…is to estimate the mean vector θ = ( θ 1 , . . . , θ n ) under squared-error loss k ˆ θ − θ k 2 , where k · k is the usual ℓ 2 -norm on R n (e.g., Abramovich et al. 2006; Brown and Greenshtein 2009; Castillo and van der Vaart 2012; Donoho and Johnstone 1994; Donoho et al. 1992; Jiang and Zhang…”
• Empirical Bayesball Remixed: Empirical Bayes Methods for Longitudinal Data (2017, methodology): “Noting that Tweedie’s formula requires only knowledge of the marginal density, g, of the observed Yi’s, Brown and Greenshtein (2009) propose simply estimating g by conventional kernel methods, thereby circumventing entirely the problem of estimating the mixing distribution, F.”
• The Poisson Compound Decision Problem Revisited (2010, methodology): “We will follow the approach of Brown and Greenshtein (2009).”