Global-local shrinkage priors have been recognized as a useful class of priors that can strongly shrink small signals toward prior means while keeping large signals unshrunk. Although such priors have been extensively discussed under Gaussian responses, in practice, we often encounter count responses. Previous contributions on global-local shrinkage priors cannot be readily applied to count data. In this paper, we discuss global-local shrinkage priors for analyzing a sequence of counts. We provide sufficient conditions under which the posterior mean is unshrunk for very large signals, known as the tail robustness property. Then, we propose tractable priors to satisfy those conditions approximately or exactly and develop a custom posterior computation algorithm for Bayesian inference without tuning parameters. We demonstrate the proposed methods through simulation studies and an application to a real dataset.
Linear regression with the classical normality assumption for the error distribution may lead to an undesirable posterior inference of regression coefficients due to the potential outliers. This paper considers the finite mixture of two components with thin and heavy tails as the error distribution, which has been routinely employed in applied statistics. For the heavily-tailed component, we introduce the novel class of distributions; their densities are log-regularly varying and have heavier tails than those of Cauchy distribution, yet they are expressed as a scale mixture of normal distributions and enable the efficient posterior inference by Gibbs sampler. We prove the robustness to outliers of the posterior distributions under the proposed models with a minimal set of assumptions, which justifies the use of shrinkage priors with unbounded densities for the high-dimensional coefficient vector in the presence of outliers. The extensive comparison with the existing methods via simulation study shows the improved performance of our model in point and interval estimation, as well as its computational efficiency. Further, we confirm the posterior robustness of our method in the empirical study with the shrinkage priors for regression coefficients.
Network meta‐analysis has been an essential methodology of systematic reviews for comparative effectiveness research. The restricted maximum likelihood (REML) method is one of the current standard inference methods for multivariate, contrast‐based meta‐analysis models, but recent studies have revealed the resultant confidence intervals of average treatment effect parameters in random‐effects models can seriously underestimate statistical errors; that is, the actual coverage probability of a true parameter cannot retain the nominal level (e.g., 95%). In this article, we provided improved inference methods for the network meta‐analysis and meta‐regression models using higher‐order asymptotic approximations based on the approach of Kenward and Roger (Biometrics 1997;53:983–997). We provided two corrected covariance matrix estimators for the REML estimator and improved approximations for its sample distribution using a t‐distribution with adequate degrees of freedom. All of the proposed procedures can be implemented using only simple matrix calculations. In simulation studies under various settings, the REML‐based Wald‐type confidence intervals seriously underestimated the statistical errors, especially in cases of small numbers of trials meta‐analyzed. By contrast, the proposed Kenward‐Roger–type inference methods consistently showed accurate coverage properties under all the settings considered in our experiments. We also illustrated the effectiveness of the proposed methods through applications to two real network meta‐analysis datasets.
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