We consider inference for a semiparametric stochastic mixed model for longitudinal data. This model uses parametric fixed effects to represent the covariate effects and an arbitrary smooth function to model the time effect. The within-subject correlation is modeled using random effects and a stationary or nonstationary stochastic process. We derive maximum penalized likelihood estimators of the regression coefficients and the nonparametric function. The resulting estimator of the nonparametric function is a smoothing spline. Frequentist and Bayesian inference on these model components are proposed and compared. Restricted maximum likelihood is used to estimate the smoothing parameter and the variance components simultaneously. We show that estimation of all model components of interest can proceed by fitting a modified linear mixed model. The proposed method is illustrated by analyzing a hormone data set and its performance is evaluated through simulations. KEY WORDS: Correlated data; ...

This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, double-exponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But the horseshoe enjoys a number of advantages over existing approaches, including its robustness, its adaptivity to different sparsity patterns, and its analytical tractability. We prove two theorems that formally characterize both the horseshoe’s adeptness at large outlying signals, and its super-efficient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using a combination of real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full Bayesian model-averaging approach using a discrete mixture prior to model signals and noise.

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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. 1. Introduction. 1.1. The model and the problems. Consider the block multivariate normal situation (sometimes called the “matrix of means problem”) specified by the following hierarchical Bayesian model:

This paper presents a general, fully Bayesian framework for sparse supervised-learning problems based on the horseshoe prior. The horseshoe prior is a member of the family of multivariate scale mixtures of normals, and is therefore closely related to widely used approaches for sparse Bayesian learning, including, among others, Laplacian priors (e.g. the LASSO) and Student-t priors (e.g. the relevance vector machine). The advantages of the horseshoe are its robustness at handling unknown sparsity and large outlying signals. These properties are justified theoretically via a representation theorem and accompanied by comprehensive empirical experiments that compare its performance to benchmark alternatives. 1

We use Lévy processes to generate joint prior distributions for a location parameter β = (β1,..., βp) as p grows large. This approach, which generalizes normal scale-mixture priors to an infinite-dimensional setting, has a number of connections with mathematical finance and Bayesian nonparametrics. We argue that it provides an intuitive framework for generating new regularization penalties and shrinkage rules; for performing asymptotic analysis on existing models; and for simplifying proofs of some classic results on normal scale mixtures.

This paper introduces an approach to estimation in possibly sparse data sets using shrinkage priors based upon the class of hypergeometric-beta distributions. These widely applicable priors turn out to be a four-parameter generalization of the beta family, and are pseudo-conjugate: they cannot themselves be expressed in closed form, but they do yield tractable moments and marginal likelihoods when used as priors for the mean of a normal distribution. These priors are useful in situations where standard priors are inappropriate or ill-behaved. Non-Bayesians will find these priors useful for generating easily computable shrinkage estimators that have excellent risk properties. Bayesians will find them useful for generating computationally tractable priors for a variance parameter. We illustrate the use of these priors on a variety of global and local shrinkage problems, and we prove a theorem that characterizes their risk proprieties when used for estimation of a normal mean under a quadratic loss function.

Let y = Aβ + ε, where y is an N × 1 vector of observations, β is a p×1 vector of unknown regression coefficients, A is an N × p design matrix and ε is a spherically symmetric error term with unknown scale parameter σ. We consider estimation of β under general quadratic loss functions, and, in particular, extend the work of Strawderman

Since Stein’s original proposal in 1962, a series of papers have constructed confidence regions of smaller volume than the standard spheres for the mean vector of a multivariate normal distribution. A general approach to this problem is developed here, and used to calculate a lower bound on the attainable volume. Bayes and fiducial methods are involved in the calculation. Scheffe-type problems are used to show that low volume by itself does not guarantee favorable inferential properties. Key Words intervals James-Stein estimator, Fisher-von Mises distribution, non-central chi, Scheffe1. Introduction Stein (1962) conjectured and heuristically demonstrated confidence regions for a multivariate normal mean vector having everywhere smaller volume than the standard ones. A variety of ingenious constructions has verified Stein’s conjecture, including those of Faith (1976), Berger (1980), Casella and Hwang (1983), Tseng and Brown (1997), and Samworth (2005). This paper concerns a general approach to multivariate normal mean

The paper is devoted to the problem of incorporating prior information in the regression analysis. Some indices of uncertainty of the prior knowledge are proposed and their usefulness is studied. To incorporate prior information together with its uncertainty into regression estimation some coefficients of uncertainty are introduced as well. Performance of estimators based upon proposed descriptions of uncertainty is examined via computer simulations. 1