Measurement error reduction using weighted average method for repeated measurements from heterogeneous instruments

Considering a situation in which observations are made by several instruments over time, researchers are interested in estimating the true quantity at a particular time point. Assuming observations made by different instruments at the same time point have the same time-dependent mean but different variances, one approach is to estimate the true quantity or the time-dependent mean by simply averaging the observations at a time point. However, since the variability of each instrument may be very different, the unweighted average can be improved by a weighted average with weights estimated from the longitudinal measurements produced by those instruments. Such a weighted average estimator is studied in this article. It is shown to be unbiased for the true quantity. Its variance is asymptotically smaller than that of any observation and also achieves the minimum variance among all the weighted estimates with known weights. The proposed method is illustrated by a real data example. The method is not restricted to the time-dependent situation. The general setting where this method can be applied is described at the end of the article.