Conditional and unconditional models in model-assisted estimation of finite population totals

The well known Godambe-Joshi lower bound for the anticipated variance of design unbiased estimators of population totals treats the auxiliary variables as constants. We extend the result to models where these variables are random and show that the generalized difference estimator using the expected values conditional on all auxiliary values is optimal. This has several implications including the fact that collecting multiple survey variables does not reduce the lower bound.