An asymptotic theory for weighted least-squares with weights estimated by replication

SUMMARY We consider a heteroscedastic linear regression model with replication. To estimate the variances, one can use the sample variances or the sample average squared errors from a regression fit. We study the large-sample properties of these weighted least-squares estimates with estimated weights when the number of replicates is small. The estimates are generally inconsistent for asymmetrically distributed data. If sample variances are used based on m replicates, the weighted least-squares estimates are inconsistent for m =2 replicates even when the data are normally distributed. With between 3 and 5 replicates, the rates of convergence are slower than the usual square root of N. With m - 6 replicates, the effect of estimating the weights is to increase variances by (m - 5)/(m -3), relative to weighted least-squares estimates with known weights.