Summary. We consider a class of ratio–product estimators for estimating a finite population mean. The asymptotically optimum estimator in the class is identified, along with its approximate mean-square error. This estimator requires prior knowledge of the parameter C=ρCy/Cx, where ρ is the correlation coefficient between the study variate y and the auxiliary variate x, and Cy and Cx are coefficients of variation of y and x respectively. If C is unknown in advance, then it can be replaced by its consistent estimate , with the resulting estimator known as an ‘estimator based on the estimating optimum’. It is shown that, to the first order of approximation, both estimators have the same mean-square error, and that they are generally more efficient than the usual ratio and product estimators.