Canadian Journal of Statistics

Ordinary and weighted least‐squares estimators

Journal Article


We propose a method of estimating the asymptotic relative efficiency (ARE) of the weighted least‐squares estimator (WLSE) with respect to the ordinary least‐squares estimator (OLSE) in a heteroscedastic linear regression model with a large number of observations but a small number of replicates at each value of the regressors. The weights used in the WLSE are the reciprocals of the (within‐group) average of squared residuals. It is shown that the OLSE is more efficient than the WLSE if the maximum number of replicates is not larger than two. The proposed estimator of the ARE is consistent as the number of observations tends to infinity. Finite‐sample performance of this estimator is examined in a simulation study. An adaptive estimator, which is asymptotically more efficient than the OLSE and the WLSE, is proposed.

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