Confidence Regions in Multivariate Calibration

Abstract

The multivariate calibration problem is considered, in which a sample of $n$ observations on vectors $\xi_{(i)}$ (of "true values") and $Y_{(i)}$ (of less accurate but more easily obtained values) are to be used to estimate the unknown $\xi$ corresponding to a future $Y$. It is assumed that $Y = BX + \varepsilon$, where $\varepsilon$ is multivariate normal and $X = h(\xi)$ for known $h$. Current methods for obtaining a confidence region $C$ for $\xi$, which consist of computing a region $R$ for $X$ and then taking $C = h^{-1}(R)$, have the disadvantage that although the region $R$ might be nicely behaved, the region $C$ need not be. An alternative method is proposed which gives a well-behaved region (corresponding to the uniformly most accurate translation-invariant region when $h$ is linear, $B$ is known and the covariance matrix of $\varepsilon$ is a known multiple of the identity). An application is given to the estimation of gestational age using ultrasound fetal bone measurements.