Hybrid confidence regions based on data depth

Summary.  We consider the general problem of constructing confidence regions for, possibly multi‐dimensional, parameters when we have available more than one approach for the construction. These approaches may be motivated by different model assumptions, different levels of approximation, different settings of tuning parameters or different Monte Carlo algorithms. Their effectiveness is often governed by different sets of conditions which are difficult to vindicate in practice. We propose two procedures for constructing hybrid confidence regions which endeavour to integrate all such individual approaches. The procedures employ the concept of data depth to calibrate the confidence region in two different ways, the first rendering its coverage error minimax and the second rendering its coverage error conservative. The resulting region reconciles in many important aspects the discrepancies between the various approaches, and is robust against misspecification of their governing conditions. Theoretical and empirical properties of our procedures are investigated in comparison with those of the constituent individual approaches.