Confidence regions in multivariate calibration (Q1098509)

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 \(Y=BX+\epsilon\), where \(\epsilon\) 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 \(\epsilon\) is a known multiple of the identity). An application is given to the estimation of gestational age using ultrasound fetal bone measurements.