William E. Strawderman

Negative Moments of Positive Random Variables

Abstract We investigate the problem of finding the expected value of functions of a random variable X of the form f(X) = (X+A)−n where X+A>0 a.s. and n is a non-negative integer. The technique is to successively integrate the probability generating function and is suggested by the well-known result that successive differentiation leads to the positive moments. The technique is applied to the problem of finding E[1/(X+A)] for the binomial and Poisson distributions.

Minimax Adaptive Generalized Ridge Regression Estimators

Abstract We consider the problem of estimating the vector of regression coefficients of a linear model using generalized ridge regression estimators where the ridge constant is chosen on the basis of the data. For general quadratic loss we produce such estimators whose risk function dominates that of the least squares procedure provided the number of regressors is at least three. We study the problem by the usual reduction to estimating the mean vector of a multivariate normal distribution. Our results on minimax estimation in this context are of independent interest.

Confidence Bands for Linear Regression with Restricted Predictor Variables

Abstract The classic procedure for constructing a confidence band around a regression function is that of Scheffe (1959), which provides a band of infinite length. This band can be unnecessarily wide if the region of interest to the experimenter is finite. A class of sets is defined that restricts the range of the predictor variables. Confidence bands for a regression function over are constructed, which, for fixed α, are narrower than the Scheffe bands. Also, a procedure is illustrated by which sets that are not in the class can be embedded in Tables are provided to facilitate the use of the procedure.

Confidence Distributions and a Unifying Framework for Meta-Analysis

This article develops a unifying framework, as well as robust meta-analysis approaches, for combining studies from independent sources. The device used in this combination is a confidence distribution (CD), which uses a distribution function, instead of a point (point estimator) or an interval (confidence interval), to estimate a parameter of interest. A CD function contains a wealth of information for inferences, and it is a useful device for combining studies from different sources. The proposed combining framework not only unifies most existing meta-analysis approaches, but also leads to development of new approaches. We illustrate in this article that this combining framework can include both the classical methods of combining p-values and modern model-based meta-analysis approaches. We also develop, under the unifying framework, two new robust meta-analysis approaches, with supporting asymptotic theory. In one approach each study size goes to infinity, and in the other approach the number of studies goes to infinity. Our theoretical development suggests that both these robust meta-analysis approaches have high breakdown points and are highly efficient for normal models. The new methodologies are applied to study-level data from publications on prophylactic use of lidocaine in heart attacks and a treatment of stomach ulcers. The robust methods performed well when data are contaminated and have realistic sample sizes and number of studies.

On estimation with balanced loss functions

Zellner ((1994) in: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics. Springer, New York, pp. 371-390), introduced the notion of a balanced loss function in the context of a general linear model to reflect both goodness of fit and precision of estimation. We study this notion from the perspective of unifying a variety of results both frequentist and Bayesian. We show in broad generality that frequentist and Bayesian results for balanced loss follow from and also imply related results for quadratic loss functions reflecting only precision of estimation. Several examples are given for normal error structures and more generally for spherically symmetric error distribution.

A new class of generalized Bayes minimax ridge regression estimators

Let y = Aβ + e, where y is an N x 1 vector of observations, β is a p x 1 vector of unknown regression coefficients, A is an N x p design matrix and e is a spherically symmetric error term with unknown scale parameter a. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman [J. Amer. Statist. Assoc. 73 (1978) 623-627] and Casella [Ann. Statist. 8 (1980) 1036-1056. J. Amer. Statist. Assoc. 80 (1985) 753-758] by finding adaptive minimax estimators (which are, under the normality assumption, also generalized Bayes) of β, which have greater numerical stability (i.e., smaller condition number) than the usual least squares estimator. In particular, we give a subclass of such estimators which, surprisingly, has a very simple form. We also show that under certain conditions the generalized Bayes minimax estimators in the normal case are also generalized Bayes and minimax in the general case of spherically symmetric errors.

Posterior propriety and admissibility of hyperpriors in normal hierarchical models

Hierarchical modeling is wonderful and here to stay, but hyperparameter priors are often chosen in a casual fashion. Unfortunately, as the number of hyperparameters grows, the effects of casual choices can multiply, leading to considerably inferior performance. As an extreme, but not uncommon, example use of the wrong hyperparameter priors can even lead to impropriety of the posterior. For exchangeable hierarchical multivariate normal models, we first determine when a standard class of hierarchical priors results in proper or improper posteriors. We next determine which elements of this class lead to admissible estimators of the mean under quadratic loss; such considerations provide one useful guideline for choice among hierarchical priors. Finally, computational issues with the resulting posterior distributions are addressed.

Confidence distribution (CD) -- distribution estimator of a parameter

The notion of confidence distribution (CD), an entirely frequentist concept, is in essence a Neymanian interpretation of Fishers Fiducial distri- bution. It contains information related to every kind of frequentist inference. In this article, a CD is viewed as a distribution estimator of a parameter. This leads naturally to consideration of the information contained in CD, com- parison of CDs and optimal CDs, and connection of the CD concept to the (profile) likelihood function. A formal development of a multiparameter CD is also presented.

A Bayesian growth and yield model for slash pine plantations

We formulate a traditional growth and yield model as a Bayes model. We attempt to introduce as few new assumptions as possible. Zellners Bayesian method of moments procedure is used, because the published model did not include any distributional assumptions. We generate predictive posterior samples for a number of stand variables using the Gibbs sampler. The means of the samples compare favorably with the predictions from the published model. In addition, our model delivers distributions of outcomes, from which it is easy to establish measures of uncertainty, such as highest posterior density regions.

Is Pitman Closeness a Reasonable Criterion

The criterion of Pitman closeness has been proposed as an alternative comparison criterion to quadratic losses and, more generally, to decision theory. However, it may lead to quite paradoxical phenomena, the most dramatic being a possible nontransitivity. The criterion takes into consideration the joint distribution of the compared estimators, but this consideration may be misleading in the selection of the “best” estimator. We show through examples that this criterion is not consistent with a decision theoretic analysis and that it should be used very cautiously, if ever.