Showing papers by "William E. Strawderman published in 2006"
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TL;DR: In this article, the authors present an expository development of Stein estimation with substantial emphasis on exact results for spherically symmetric distributions, showing that the improvement possible over the best invariant estimator via shrinkage estimation is not surprising but expected from a variety of perspectives.
Abstract: This paper presents an expository development of Stein estimation with substantial emphasis of exact results for spherically symmetric distributions. The themes of the paper are: a) that the improvement possible over the best invariant estimator via shrinkage estimation is not surprising but expected from a variety of perspectives; b) that the amount of shrinkage allowable to preserve domination over the best invariant estimator is, when properly interpreted, relatively free from the assumption of normality; and c) that the potential savings in risk are substantial when accompanied by good quality prior information.
65 citations
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TL;DR: In this paper, the authors extend the results of Stein to spherically symmetric distributions and investigate conditions under which estimators of the form X ag(X) dominate X for loss functions which are concave in ||δ- θ||2.
Abstract: This paper is primarily concerned with extending the results of Stein to spherically symmetric distributions Specifically, when X ∼f(||X - θ||2), we investigate conditions under which estimators of the form X ag(X) dominate X for loss functions ||δ- θ||2 and loss functions which are concave in ||δ- θ||2 Additionally, if the scale is unknown we investigate estimators of the location parameter of the form X aVg(X) in two different settings In the first, an estimator V of the scale is independent of X In the second, V is the sum of squared residuals in the usual canonical setting of a generalized linear model when sampling from a spherically symmetric distribution These results are also generalized to concave loss The conditions for domination of X ag(X) are typically (a) ||g||2 2∇∘g ≤0, (b) ∇∘g is superharmonic and (c) 0
60 citations
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TL;DR: In this paper, the minimax estimators of Theta whose risks are smaller than the risk of X (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss.
Abstract: For p >4 and one observation X on a p-dimensional spherically symmetric distribution, minimax estimators of Theta whose risks are smaller than the risk of X (the best invariant estimator) are found when the loss is a nondecreasing concave function of quadratic loss. For n observations X1, X2, ... Xn, we have classes of minimax estimators which are better than the usual procedures, such as the best invariant estimator, X-bar, or a maximum likelihood estimator.
54 citations
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TL;DR: In this article, a family of minimax estimators for the location parameter of a p-variate (p> or = 3) spherically symmetric unimodal (s.s.u.)distribution with respect to general quadratic loss is found.
Abstract: Families of minimax estimators are found for the location parameter of a p-variate (p> or = 3) spherically symmetric unimodal(s.s.u.)distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by L(delta,theta) = (delta - theta)'D(delta - theta) where D is a known pxp positive definite matrix, one main result, for one observation, X, on a multivariate s.s.u. distribution about theta, presents a class of minimax estimators whose risks dominate the risk of X, provided p> or = 3 and trace D > or equal 2dL where dL is the maximum eigenvalue of D. This class is given by Delta a,r(X)=(1-a(r(||X||2)/||X||2))X where 0 or = 4 and co = .96 when p=3.
50 citations
TL;DR: In this article, a Bayesian forecast of the distribution of returns by stochastic approximation is used for portfolio selection methodology using a hierarchical priors on the mean vector and covariance matrix of returns.
Abstract: This paper contributes to portfolio selection methodology using a Bayesian forecast of the distribution of returns by stochastic approximation. New hierarchical priors on the mean vector and covariance matrix of returns are derived and implemented. Comparison’s between this approach and other Bayesian methods are studied with simulations on 25 years of historical data on global stock indices. It is demonstrated that a fully hierarchical Bayes procedure produces promising results warranting more study. We carried out a numerical optimization procedure to maximize expected utility using the MCMC (Monte Carlo Markov Chain) samples from the posterior predictive distribution. This model resulted in an extra 1.5 percentage points per year in additional portfolio performance (on top of the Hierarchical Bayes model to estimate μ and Σ and use the Markowitz model), which is quite a significant empirical result. This approach applies to a large class of utility functions and models for market returns.
42 citations
TL;DR: In this paper, two expectation identities and a series of applications are presented, including exact formulas and bounds for moments, an improvement and a reversal of Jensen's inequality, linking unbiased estimation to elliptic partial differential equations, applications to decision theory and Bayesian statistics, and an application to counting matchings in graph theory.
32 citations
TL;DR: In this paper, a new class of minimax generalized Bayes estimators of the variance of a normal distribution is given under both quadratic and entropy losses, and the new class may have a noticeably larger region of substantial improvement over the usual estimator than Brewster and Zidek-type procedures.
20 citations
TL;DR: A statistical analysis for key comparisons with linear trends and multiple artefacts is proposed, extension of a previous paper for a single artefact that has the advantage that it is consistent with the no-trend case.
Abstract: A statistical analysis for key comparisons with linear trends and multiple artefacts is proposed. This is an extension of a previous paper for a single artefact. The approach has the advantage that it is consistent with the no-trend case. The uncertainties for the key comparison reference value and the degrees of equivalence are also provided. As an example, the approach is applied to key comparison CCEM–K2.
16 citations
TL;DR: For example, Zhang and Woodroofe as discussed by the authors established the attractive lower bound of 1 − 1 − α 1+ α for the coverage probability of the truncation of non-informative priors onto the restricted parameter space.
Abstract: For estimating a positive normal mean, Zhang and Woodroofe (2003) as well as Roe and Woodroofe (2000) investigate 100($1-\alpha)%$ HPD credible sets associated with priors obtained as the truncation of noninformative priors onto the restricted parameter space. Namely, they establish the attractive lower bound of $\frac{1-\alpha}{1+\alpha}$ for the frequentist coverage probability of these procedures. In this work, we establish that the lower bound of $\frac{1-\alpha}{1+\alpha}$ is applicable for a substantially more general setting with underlying distributional symmetry, and obtain various other properties. The derivations are unified and are driven by the choice of a right Haar invariant prior. Investigations of non-symmetric models are carried out and similar results are obtained. Namely, (i) we show that the lower bound $\frac{1-\alpha}{1+\alpha}$ still applies for certain types of asymmetry (or skewness), and (ii) we extend results obtained by Zhang and Woodroofe (2002) for estimating the scale parameter of a Fisher distribution; which arises in estimating the ratio of variance components in a one-way balanced random effects ANOVA. Finally, various examples illustrating the wide scope of applications are expanded upon. Examples include estimating parameters in location models and location-scale models, estimating scale parameters in scale models, estimating linear combinations of location parameters such as differences, estimating ratios of scale parameters, and problems with non-independent observations.
14 citations
TL;DR: In this paper, the authors consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter and give estimators which improve on the usual unbiased estimator.
Abstract: In this article we consider estimating a location parameter of a spherically symmetric distribution under restrictions on the parameter. First we consider a general theory for estimation on polyhedral cones which includes examples such as ordered parameters and general linear inequality restrictions. Next, we extend the theory to cones with piecewise smooth boundaries. Finally we consider shrinkage toward a closed convex set K where one has vague prior information that θ is in K but where θ is not restricted to be in K. In this latter case we give estimators which improve on the usual unbiased estimator while in the restricted parameter case we give estimators which improve on the projection onto the cone of the unbiased estimator. The class of estimators is somewhat non-standard as the nature of the constraint set may preclude weakly differentiable shrinkage functions. The technique of proof is novel in the sense that we first deduce the improvement results for the normal location problem and then extend them to the general spherically symmetric case by combining arguments about uniform distributions on the spheres, conditioning and completeness.
13 citations
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TL;DR: In this paper, the authors present an expository development of Bayesian estimation with substantial emphasis on exact results for the multivariate normal location models with respect to squared error loss, and give a coherent presentation of numerous Bayesian results (proper, generalized, empirical).
Abstract: This paper presents an expository development of Bayesian estimation with substantial emphasis on exact results for the multivariate normal location models with respect to squared error loss. From the time Stein, in 1956, showed the inadmissibility of the best invariant estimator when sampling from a multivariate normal distribution in 3 or more dimensions, there has been an outpouring of improved estimators with a Bayesian flavor, encouraged largely by the connections between Bayes estimation, admissibility and minimaxity. In this chapter, we attempt to give a coherent presentation of numerous Bayesian results (proper, generalized, empirical) for this case. Generalizations for the location parameter of multivariate normal distributions with unknown covariance matrices and general quadratic loss are also presented.
TL;DR: The methodology was developed for the case where prior delineation of unimpacted areas is not possible because of site history and a large set of ground-water monitoring measurements exists.
Abstract: A statistical methodology formulated for defining background or baseline levels of constituents of concern in groundwater is presented. The methodology was developed for the case where prior delineation of unimpacted areas is not possible because of site history and a large set of groundwater monitoring measurements exists. Consideration was given to spatial and temporal trends, outliers, and final segregation of wells into impacted or unimpacted categories to develop probability distributions and summary statistics for each constituent evaluated. The formulated approaches were applied to groundwater monitoring data for the U.S. Department of Energy Savannah River Site facility, and results for four representative constituents (aluminum, arsenic, mercury, and tritium) are discussed.