Shrinkage estimation and the use of additional information when calibrating in the presence of random effects

Let x denote a precise measurement of a quantity and Y an inexact measurement, which is, however, less expensive or more easily obtained than x. We have available a calibration set comprising clustered sets of (x,Y) observations, obtained from different sampling units. At the prediction step, we will only observe Y for a new unit, and we wish to estimate the corresponding unknown x, which we denote by ξ. This problem has been treated under the assumption that x and Y are linearly related. Here, we expand on those results in three directions: First, we show that if we center ξ about a known value c, for example, the mean x-value of the calibration set, then the proposed estimator now shrinks to c. Second, we examine in detail the performance of the estimator, which was proposed when one or more (x,Y) observations can be obtained for the new subject. Third, we compare the Fieller-like confidence intervals, previously proposed, with t-like intervals based on asymptotic moments of the point estimate. We illustrate and evaluate our procedures in the context of a data set of true bladder-volumes (x) and ultrasound measurements (Y).