Can the sample variance estimator be improved by using a covariate?

Variance components are quantities of central interest in many applications, for example in cultivar yield stability analysis and in the analysis of measurement errors. In some applications, the feasible sample size is rather limited leading to estimates of variance components which are subject to considerable sampling variation. For example, new crop cultivars are tested for only a few years before release to the market, so the sample size for the year variance component is small. Similarly, testing a new measurement instrument for some chemical compound may be costly, allowing only a limited number of replications. This paper investigates the potential for improving the usual sample variance estimator by exploiting covariate information. In a cultivar trial, yield data may be available for only a few environments, while meteorological data or data on a standard cultivar has been recorded for a very large number of environments. Likewise, in the analysis of measurement errors, there may be long-term data on a standard measurement procedure that can be used as a covariate to improve the variance estimate for a new instrument. It is shown in this paper that the gain in accuracy by a covariate can be considerable, provided there is sufficient correlation between the covariate and the variable of interest