Exact optimal and adaptive inference in regression models under heteroskedasticity and non-normality of unknown forms

In this paper, we derive simple point-optimal sign-based tests in the context of linear and nonlinear regression models with fixed regressors. These tests are exact, distribution-free, robust against heteroskedasticity of unknown form, and they may be inverted to obtain confidence regions for the vector of unknown parameters. Since the point-optimal sign tests depend on the alternative hypothesis, we propose an adaptive approach based on split-sample techniques in order to choose an alternative such that the power of point-optimal sign tests is close to the power envelope. The simulation results show that when using approximately 10% of sample to estimate the alternative and the rest to calculate the test statistic, the power of point-optimal sign test is typically close to the power envelope. We present a Monte Carlo study to assess the performance of the proposed “quasi”-point-optimal sign test by comparing its size and power to those of some common tests which are supposed to be robust against heteroskedasticity. The results show that our procedures are superior

Prior distributions for objective Bayesian analysis

We provide a review of prior distributions for objective Bayesian analysis. We start by examining some foundational issues and then organize our exposition into priors for: i) estimation or prediction; ii) model selection; iii) highdimensional models. With regard to i), we present some basic notions, and then move to more recent contributions on discrete parameter space, hierarchical models, nonparametric models, and penalizing complexity priors. Point ii) is the focus of this paper: it discusses principles for objective Bayesian model comparison, and singles out some major concepts for building priors, which are subsequently illustrated in some detail for the classic problem of variable selection in normal linear models. We also present some recent contributions in the area of objective priors on model space.With regard to point iii) we only provide a short summary of some default priors for high-dimensional models, a rapidly growing area of research

Good, great, or lucky? Screening for firms with sustained superior performance using heavy-tailed priors

This paper examines historical patterns of ROA (return on assets) for a cohort of 53,038 publicly traded firms across 93 countries, measured over the past 45 years. Our goal is to screen for firms whose ROA trajectories suggest that they have systematically outperformed their peer groups over time. Such a project faces at least three statistical difficulties: adjustment for relevant covariates, massive multiplicity, and longitudinal dependence. We conclude that, once these difficulties are taken into account, demonstrably superior performance appears to be quite rare. We compare our findings with other recent management studies on the same subject, and with the popular literature on corporate success. Our methodological contribution is to propose a new class of priors for use in large-scale simultaneous testing. These priors are based on the hypergeometric inverted-beta family, and have two main attractive features: heavy tails and computational tractability. The family is a four-parameter generalization of the normal/inverted-beta prior, and is the natural conjugate prior for shrinkage coefficients in a hierarchical normal model. Our results emphasize the usefulness of these heavy-tailed priors in large multiple-testing problems, as they have a mild rate of tail decay in the marginal likelihood $m(y)$---a property long recognized to be important in testing.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS512 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian semiparametric analysis for two-phase studies of gene-environment interaction

The two-phase sampling design is a cost-efficient way of collecting expensive covariate information on a judiciously selected subsample. It is natural to apply such a strategy for collecting genetic data in a subsample enriched for exposure to environmental factors for gene-environment interaction (G x E) analysis. In this paper, we consider two-phase studies of G x E interaction where phase I data are available on exposure, covariates and disease status. Stratified sampling is done to prioritize individuals for genotyping at phase II conditional on disease and exposure. We consider a Bayesian analysis based on the joint retrospective likelihood of phases I and II data. We address several important statistical issues: (i) we consider a model with multiple genes, environmental factors and their pairwise interactions. We employ a Bayesian variable selection algorithm to reduce the dimensionality of this potentially high-dimensional model; (ii) we use the assumption of gene-gene and gene-environment independence to trade off between bias and efficiency for estimating the interaction parameters through use of hierarchical priors reflecting this assumption; (iii) we posit a flexible model for the joint distribution of the phase I categorical variables using the nonparametric Bayes construction of Dunson and Xing [J. Amer. Statist. Assoc. 104 (2009) 1042-1051].Comment: Published in at http://dx.doi.org/10.1214/12-AOAS599 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org