Regional hydrologic analysis: Ordinary and generalized least squares revisited

Generalized least squares (GLS) regional regression procedures have been developed for estimating river flow quantiles. A widely used GLS procedure employs simplified model error structure and average covariances when constructing an approximate residual error covariance matrix. This paper compares that GLS estimator (β(GLS)/(MC)), an idealized GLS estimator (β(GLS)/(E)) based on the simplifying assumptions of β(GLS)/(MC) with true underlying statistics in a region, the best possible GLS estimator (β(GLS/(T)) obtained using the true residual error covariance matrix, and the ordinary least squares estimator (β(OLS)/(T)). Useful analytic expressions are developed for the variance of β(GLS)/(T), β(GLS)/(E), and β(OLS)/(T). For previously examined cases the average sampling mean square error (mse(s)) of β(GLS)/(E) was the same as the mse(s) of β(GLS)/(T), and the mse(s) of β(GLS)/(MC) usually was larger than the mse(s) of both β(GLS)/(E) and β(GLS)/(T). The loss in efficiency of β(GLS)/(MC) was mostly due to estimating streamflow statistics employed in the construction of the residual error covariance matrix rather than the simplifying assumptions in presently employed GLS estimators. The new analytic expression were used to compare the performance of the OLS and GLS estimators for new cases representing greater model variability across sites as well as the effect return period has on the estimators' relative performance. For a more heteroscedastic model error variance and larger return periods, some increase in the mse(s) of β(GLS)/(E) relative to the mse(s) of β(GLS)/(T) was observed.