Tolerance regions and multiple-use confidence regions in multivariate calibration

Let y(i) similar to N(Bx(i), Sigma), i = 1, 2,..., N, and y similar to N(B theta, Sigma) be independent multivariate observations, where the x(i)'s ar e known vectors, B, theta and Sigma are unknown, B being a positive definit e matrix. The calibration problem deals with statistical inference concerni ng theta and the problem that we have addressed is the construction of conf idence regions. In this article, we have constructed a region for theta bas ed on a criterion similar to that satisfied by a tolerance region. The situ ation where theta is possibly a nonlinear function, say h(xi), of fewer unk nown parameters denoted by the vector xi, is also considered. The problem a ddressed in this context is the construction of a region for xi. The numeri cal computations required for the practical implementation of our region ar e explained in detail and illustrated using an example. Limited numerical r esults indicate that our regions satisfy the coverage probability requireme nts of multiple-use confidence regions.