Regression analysis of data from a cluster sample

For the case of different regression relationships in different subgroups of a finite population, only part of which are sampled, the extended least squares estimator of any weighted average of the distinct coefficients is derived, under assumptions relating only to the first two moments of the distribution of the coefficients. Under these assumptions, the estimator is shown to be the best linear ξ-unbiased estimator, while under further distributional assumptions, it is also the Bayesian estimator for a quadratic loss function. For the case of unknown variances a method for estimating them from the sample is proposed. The empirical estimator thus obtained is shown to perform well by a simulation comparison with the optimal estimator and with other proposed empirical estimators.