Simultaneous Confidence Regions and Weighted Hypotheses on Parameter Arrays

Testing weighted hypotheses simultaneously for a parameter vector has been actively studied in the literature, where the weights encompass information on the importance of the parameters. However, in recent applications of big data analytics for multiple testing on n hypotheses, we are often confronted with the problem of simultaneous inference on a parameter matrix, not a parameter vector. For instance, in the evaluation of overall system reliability, when each subsystem contains k multiple components, the control of global confidence level for the evaluation on system reliability necessitates simultaneous inference on all related parameters in the array of a k× n matrix, where weights are assigned on the basis of the subsystem workloads. So far as we know, there is no publication addressing weighted confidence sets for a parameter matrix. In this paper, we propose a confidence algorithm that generates confidence regions for simultaneous estimation on the parameter array. The new method utilizes a random partition in conjunction with weight assignments to justify for multiplicity. After theoretical derivations, we present simulation studies that cast new lights on intrinsic relationships among coverage probabilities, power performance, and hypothesis weights for multivariate simultaneous confidence sets. For illustration purposes, the new method is applied to analyze factors impacting the taste of red and white wine in a recent study.