Asymptotic Theory for Weighted Least Squares with Weights Estimated by Replication

摘要

This document considers 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. The authors 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 > or = 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. (KR)