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William E. Strawderman

Publications and source records attributed to William E. Strawderman.

At least 19 recordsLinked to original sources
arXiv2026

Proper Bayes minimax multiple shrinkage estimation

For the canonical problem of estimating a multivariate normal mean under squared error loss, we demonstrate, for the first time, the existence of proper Bayes minimax multiple shrinkage estimators by introducing a general approach for their explicit construction. As opposed to minimax shrinkage estimators that shrink towards a single prespecified target, minimax multiple shrinkage estimators adaptively shrink towards the more promising of a set of prespecified targets, substantially increasing the region of potential risk reduction while maintaining the protection of always being at least as good as the maximum likelihood estimator. These estimators are particularly useful in practice as they address the challenge of selecting a minimax shrinkage estimator when prior information suggests more than one viable shrinkage target to choose from. In contrast to previous formal Bayes minimax multiple shrinkage estimators, which were built on mixtures of superharmonic marginals, these proper Bayes minimax multiple shrinkage estimators are obtained via mixtures of square-root superharmonic marginals. Examples of such proper Bayes minimax multiple shrinkage estimators include an adaptive convex combination of the rescaled Strawderman shrinkage estimators.

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arXiv2025

Constructing Bayes Minimax Estimators through Integral Transformations

The problem of Bayes minimax estimation for the mean of a multivariate normal distribution under quadratic loss has attracted significant attention recently. These estimators have the advantageous property of being admissible, similar to Bayes procedures, while also providing the conservative risk guarantees typical of frequentist methods. This paper demonstrates that Bayes minimax estimators can be derived using integral transformation techniques, specifically through the \( I \)-transform and the Laplace transform, as long as appropriate spherical priors are selected. Several illustrative examples are included to highlight the effectiveness of the proposed approach.

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arXiv2024

Double shrinkage priors for a normal mean matrix

We consider estimation of a normal mean matrix under the Frobenius loss. Motivated by the Efron--Morris estimator, a generalization of Stein's prior has been recently developed, which is superharmonic and shrinks the singular values towards zero. The generalized Bayes estimator with respect to this prior is minimax and dominates the maximum likelihood estimator. However, here we show that it is inadmissible by using Brown's condition. Then, we develop two types of priors that provide improved generalized Bayes estimators and examine their performance numerically. The proposed priors attain risk reduction by adding scalar shrinkage or column-wise shrinkage to singular value shrinkage. Parallel results for Bayesian predictive densities are also given.

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arXiv2021

Estimation under matrix quadratic loss and matrix superharmonicity

We investigate estimation of a normal mean matrix under the matrix quadratic loss. Improved estimation under the matrix quadratic loss implies improved estimation of any linear combination of the columns. First, an unbiased estimate of risk is derived and the Efron--Morris estimator is shown to be minimax. Next, a notion of \textit{matrix superharmonicity} for matrix-variate functions is introduced and shown to have analogous properties with usual superharmonic functions, which may be of independent interest. Then, we show that the generalized Bayes estimator with respect to a matrix superharmonic prior is minimax. We also provide a class of matrix superharmonic priors that includes the previously proposed generalization of Stein's prior. Numerical results demonstrate that matrix superharmonic priors work well for low rank matrices.

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arXiv2021

On admissible estimation of a mean vector when the scale is unknown

We consider admissibility of generalized Bayes estimators of the mean of a multivariate normal distribution when the scale is unknown under quadratic loss. The priors considered put the improper invariant prior on the scale while the prior on the mean has a hierarchical normal structure conditional on the scale. This conditional hierarchical prior is essentially that of Maruyama and Strawderman (2021, Biometrika) (MS21) which is indexed by a hyperparameter $a$. In that paper $a$ is chosen so this conditional prior is proper which corresponds to $a>-1$. This paper extends MS21 by considering improper conditional priors with $a$ in the closed interval $[-2, -1]$, and establishing admissibility for such $a$. The authors, in Maruyama and Strawderman (2017, JMVA), have earlier shown that such conditional priors with $a < -2$ lead to inadmissible estimators. This paper therefore completes the determination of admissibility/inadmissibility for this class of priors. It establishes the the boundary as $a = -2$, with admissibility holding for $a\geq -2$ and inadmissibility for $a < -2$. This boundary corresponds exactly to that in the known scale case for these conditional priors, and which follows from Brown (1971, AOMS). As a notable benefit of this enlargement of the class of admissible generalized Bayes estimators, we give admissible and minimax estimators in all dimensions greater than $2$ as opposed to MS21 which required the dimension to be greater than $4$. In one particularly interesting special case, we establish that the joint Stein prior for the unknown scale case leads to a minimax admissible estimator in all dimensions greater than $2$.

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arXiv2019

On shrinkage estimation for balanced loss functions

The estimation of a multivariate mean $θ$ is considered under natural modifications of balanced loss function of the form: (i) $ω\, ρ(\|δ-δ_0\|^2) + (1-ω) \, ρ(\|δ-θ\|^2) $, and (ii) $\ell \left( ω\, \|δ-δ_0\|^2 + (1-ω) \, \|δ-θ\|^2 \right)\,$, where $δ_0$ is a target estimator of $γ(θ)$. After briefly reviewing known results for original balanced loss with identity $ρ$ or $\ell$, we provide, for increasing and concave $ρ$ and $\ell$ which also satisfy a completely monotone property, Baranchik-type estimators of $θ$ which dominate the benchmark $δ_0(X)=X$ for $X$ either distributed as multivariate normal or as a scale mixture of normals. Implications are given with respect to model robustness and simultaneous dominance with respect to either $ρ$ or $\ell

math.ST↗
arXiv2019

Predictive density estimation under the Wasserstein loss

We investigate predictive density estimation under the $L^2$ Wasserstein loss for location families and location-scale families. We show that plug-in densities form a complete class and that the Bayesian predictive density is given by the plug-in density with the posterior mean of the location and scale parameters. We provide Bayesian predictive densities that dominate the best equivariant one in normal models.

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arXiv2018

A Gaussian sequence approach for proving minimaxity: A Review

This paper reviews minimax best equivariant estimation in these invariant estimation problems: a location parameter, a scale parameter and a (Wishart) covariance matrix. We briefly review development of the best equivariant estimator as a generalized Bayes estimator relative to right invariant Haar measure in each case. Then we prove minimaxity of the best equivariant procedure by giving a least favorable prior sequence based on non-truncated Gaussian distributions. The results in this paper are all known, but we bring a fresh and somewhat unified approach by using, in contrast to most proofs in the literature, a smooth sequence of non truncated priors. This approach leads to some simplifications in the minimaxity proofs.

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arXiv2018

On efficient prediction and predictive density estimation for spherically symmetric models

Let $X,U,Y$ be spherically symmetric distributed having density $$η^{d +k/2} \, f\left(η(\|x-θ|^2+ \|u\|^2 + \|y-cθ\|^2 ) \right)\,,$$ with unknown parameters $θ\in \mathbb{R}^d$ and $η>0$, and with known density $f$ and constant $c >0$. Based on observing $X=x,U=u$, we consider the problem of obtaining a predictive density $\hat{q}(y;x,u)$ for $Y$ as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant density $\hat{q}_{mre}$, which is Generalized Bayes with respect to the prior $π(θ, η) = η^{-1}$. For $d \geq 3$, we obtain improvements on $\hat{q}_{mre}$, and further show that the dominance holds simultaneously for all $f$ subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior $π_h(θ, η) =η^{-1} \|θ\|^{2-d}$ dominates $\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.

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arXiv2017

Admissible Bayes equivariant estimation of location vectors for spherically symmetric distributions with unknown scale

This paper investigates estimation of the mean vector under invariant quadratic loss for a spherically symmetric location family with a residual vector with density of the form $ f(x,u)=η^{(p+n)/2}f(η\{\|x-θ\|^2+\|u\|^2\}) $, where $η$ is unknown. We show that the natural estimator $x$ is admissible for $p=1,2$. Also, for $p\geq 3$, we find classes of generalized Bayes estimators that are admissible within the class of equivariant estimators of the form $\{1-ξ(x/\|u\|)\}x$. In the Gaussian case, a variant of the James--Stein estimator, $[1-\{(p-2)/(n+2)\}/\{\|x\|^2/\|u\|^2+(p-2)/(n+2)+1\}]x$, which dominates the natural estimator $x$, is also admissible within this class. We also study the related regression model.

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arXiv2016

A sharp boundary for SURE-based admissibility for the Normal means problem under unknown scale

We consider quasi-admissibility/inadmissibility of Stein-type shrinkage estimators of the mean of a multivariate normal distribution with covariance matrix an unknown multiple of the identity. Quasi-admissibility/inadmissibility is defined in terms of non-existence/existence of a solution to a differential inequality based on Stein's unbiased risk estimate (SURE). We find a sharp boundary between quasi-admissible and quasi-inadmissible estimators related to the optimal James-Stein estimator. We also find a class of priors related to the Strawderman class in the known variance case where the boundary between quasi-admissibility and quasi-inadmissibility corresponds to the boundary between admissibility and inadmissibility in the known variance case. Additionally, we also briefly consider generalization to the case of general spherically symmetric distributions with a residual vector.

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arXiv2014

On Predictive Density Estimation for Location Families under Integrated $L_2$ and $L_1$ Losses

Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequentist risk of such predictive densities with integrated $L_2$ and $L_1$ losses. Our findings relate to a $p-$variate spherically symmetric observable $X \sim p_X(\|x-μ\|^2)$ and the objective of estimating the density of $Y \sim q_Y(\|y-μ\|^2)$ based on $X$. For $L_2$ loss, we describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three of more dimensions (i.e., $p \geq 3$), we show that the MRE estimator is inadmissible under $L_2$ loss and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of $\|\hatμ-μ\|^2$. We also study the phenomenon of improvement on the plug-in density estimator of the form $q_Y(\|y-aX\|^2)\,, 0 1$, showing in some cases, inevitably for large enough $p$, that all choices $c>1$ are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for $p \geq 3$. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with $p \geq 4$ under $L_1$ loss.

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arXiv2013

Dominance properties of constrained Bayes and empirical Bayes estimators

This paper studies decision theoretic properties of benchmarked estimators which are of some importance in small area estimation problems. Benchmarking is intended to improve certain aggregate properties (such as study-wide averages) when model based estimates have been applied to individual small areas. We study decision-theoretic properties of such estimators by reducing the problem to one of studying these problems in a related derived problem. For certain such problems, we show that unconstrained solutions in the original (unbenchmarked) problem give unconstrained Bayes and improved estimators which automatically satisfy the benchmark constraint. Also, dominance properties of constrained empirical Bayes estimators are shown in the Fay-Herriot model, a frequently used model in small area estimation.

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arXiv2013

Improved robust Bayes estimators of the error variance in linear models

We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over the entire class of scale mixture of Gaussian distributions.

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arXiv2013

Robust Bayesian variable selection with sub-harmonic priors

This paper studies Bayesian variable selection in linear models with general spherically symmetric error distributions. We propose sub-harmonic priors which arise as a class of mixtures of Zellner's g-priors for which the Bayes factors are independent of the underlying error distribution, as long as it is in the spherically symmetric class. Because of this invariance to spherically symmetric error distribution, we refer to our method as a robust Bayesian variable selection method. We demonstrate that our Bayes factors have model selection consistency and are coherent. We also develop Laplace approximations to Bayes factors for a number of recently studied mixtures of g-priors that have recently appeared in the literature (including our own) for Gaussian errors. These approximations, in each case, are given by the Gaussian Bayes factor based on BIC times a simple rational function of the prior's hyper-parameters and the R^2's for the respective models. We also extend model selection consistency for several g-prior based Bayes factor methods for Gaussian errors to the entire class of spherically symmetric error distributions. Additionally we demonstrate that our class of sub-harmonic priors are the only ones within a large class of mixtures of g-priors studied in the literature which are robust in our sense. A simulation study and an analysis of two real data sets indicates good performance of our robust Bayes factors relative to BIC and to other mixture of g-prior based methods.

stat.ME↗
arXiv2012

On Bayesian credible sets in restricted parameter space problems and lower bounds for frequentist coverage

For estimating a lower bounded parametric function in the framework of Marchand and Strawderman (2006), we provide through a unified approach a class of Bayesian confidence intervals with credibility $1-α$ and frequentist coverage probability bounded below by $\frac{1-α}{1+α}$. In cases where the underlying pivotal distribution is symmetric, the findings represent extensions with respect to the specification of the credible set achieved through the choice of a {\it spending function}, and include Marchand and Strawderman's HPD procedure result. For non-symmetric cases, the determination of a such a class of Bayesian credible sets fills a gap in the literature and includes an "equal-tails" modification of the HPD procedure. Several examples are presented demonstrating wide applicability.

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arXiv2012

A unified minimax result for restricted parameter spaces

We provide a development that unifies, simplifies and extends considerably a number of minimax results in the restricted parameter space literature. Various applications follow, such as that of estimating location or scale parameters under a lower (or upper) bound restriction, location parameter vectors restricted to a polyhedral cone, scale parameters subject to restricted ratios or products, linear combinations of restricted location parameters, location parameters bounded to an interval with unknown scale, quantiles for location-scale families with parametric restrictions and restricted covariance matrices.

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