An inference procedure for order parameters utilizing confidence distribution random variables

Meta-analysis is a valuable tool for combining information from independent studies. However, most common meta-analysis techniques rely on distributional assumptions that are difficult, if not impossible, to verify. For instance, in the commonly used fixed-effects and random-effects models, we take for granted that the underlying study parameters are either exactly the same across individual studies or that they are samples from a single (often normal) distribution. In this paper, we present a new framework for summarizing information obtained from multiple studies and make inference that is not dependent on any distributional assumption for the study-level treatment effects. This development leads to an inference problem of constructing confidence intervals or hypothesis tests for order parameters of the treatment effects, including the extrema of parameters, (e.g., max{θ1, ..., θK}). Such an inference problem is considered in the literature as one of “the existing problems where standard bootstrap estimators are not consistent and where alternative approaches also face significant challenges” (Hall and Miller, 2010). Based on recent developments on confidence distributions, we propose a new resampling method to deal with the inference problem for the extrema of the parameters and also, more generally, for any order parameters. This new resampling method can be viewed as an extension of the well-studied and widely-used bootstrap method, but it enjoys a more flexible interpretation and manipulation. We provide a large sample theoretical support for the proposed method. We also explore the theoretical performance of both the standard bootstrap and proposed method, especially in the presence of ties and near ties among the θi’s. We discuss tuning methods that provide good finite-sample performance, and show that our results compare favorably to comparable estimators, including the standard bootstrap. Finally, we illustrate our method with