New multistage formulations of minimum risk fixed-size confidence region (MRFSCR) problems for estimating a multivariate normal mean with illustrations, simulations and data analysis

Classical fixed-size confidence region (FSCR) methods for a multinormal mean tend to prefix the maximum diameter of a confidence region. However, we feel that the size of a confidence region should incorporate the quality of available data instead of fixing it arbitrarily. We begin with a new FSCR method for the mean of a multinormal population having a dispersion matrix σ2H with H known, but allow the region’s size to involve σ. On a separate track, the minimum risk point estimation (MRPE) problems have included an explicit loss function balancing estimation error and the cost of data. The absence of an explicit loss function within FSCR methods motivated us to formulate a novel minimum risk fixed size confidence region (MRFSCR) problem for a multinormal mean where the size of the confidence region is allowed to involve σ. We first build a unified structure under multistage sampling to construct MRFSCR with asymptotic first-order (f.o.) and second-order (s.o.) properties. We supplement the general theory and methodology by appealing to (i) a number of practical multistage strategies as special cases; (ii) extensive analyses of simulated data; and (iii) illustrations using two interesting real data.