Abstract

In traditional fixed-size confidence region (FSCR) estimation, ellipsoidal confidence regions of fixed diameter are often constructed arbitrarily for the mean vector of a multivariate normal distribution. However, it is more appropriate to determine the size of the confidence region based on the quality of available data. This paper introduces a novel formulation of FSCR for the multinormal mean, specifically considering the dispersion matrix in the form of $\sigma^2H$, where the size of the confidence region is expressed as a function of $\sigma$. Unlike minimum risk point estimation (MRPE) problems, which incorporate explicit loss functions to balance estimation error and sampling cost, FSCR lacks such considerations. Motivated by this, a new framework is proposed: minimum risk fixed-size confidence region (MRFSCR) for the mean vector in multivariate normal distributions. Additionally, a unified structure for multistage sampling strategies to construct MRFSCRs is presented, with demonstrated asymptotic first-order and second-order properties. Illustrations and analyses using simulated data complement the theoretical and methodological discussions.