Tolerance regions and multiple-use confidence regions in multivariate calibration

Abstract

Let $\mathrm{y}~ N(Bx_i,\Sigma),i=1,2\ldots,N$, and $\mathrm{y}~N(B\theta, \Sigma)$ be independent multivariate observations, where the $x_i$'s are known vectors, $B,\theta$ and $\Sigma$ are unknown, $\Sigma$ being a positive definite matrix. The calibration problem deals with statistical inference concerning $\theta$ and the problem that we have addressed is the construction of confidence regions. In this article, we have constructed a region for $\theta$ based on a criterion similar to that satisfied by a tolerance region. The situation where $\theta$ is possibly a nonlinear function, say $\mathrm{h}(\xi)$ of fewer unknown parameters denoted by the vector $(\xi)$, is also considered. The problem addressed in this context is the construction of a region for $\xi$. The numerical computations required for the practical implementation of our region are explained in detail and illustrated using an example. Limited numerical results indicate that our regions satisfy the coverage probability requirements of multiple­use confidence regions.