A method is given which provides, under conditions satisfied by many common distributions, rules for sampling in two stages so as to obtain an unbiased estimator of a given parameter, having variance equal to, or not exceeding, a prescribed bound. The method is applied to estimation of the means of binomial, Poisson, and hypergeometric distributions; scale-parameters in general and of the Gamma distribution in particular; the variance of a normal distribution; and a component of variance. The use of such estimators to achieve homoscedasticity is discussed. Optimum sampling rules are discussed for some of these estimators, and some tables are given to facilitate their use. The efficiency of the method is shown to be high in many cases.