In sampling from a finite population, the nonexistence of a uniformly minimum variance unbiased estimator for the mean μ has been demonstrated by Godambe [3], and the inadmissibility of the sample mean as an estimator for μ, when sampling is with replacement and equal probabilities, has been proved by Des Raj and Khamis [2] and by Basu [1]. In this paper, the problem of unbiased linear estimation of μ with minimum variance is considered for a very general scheme of sampling. An admissible estimator is obtained, together with a complete class of estimators. It is shown further that, for a somewhat restricted sampling scheme, amongst estimators with variance proportional to σ², there does exist a best estimator which, in the case of sampling with replacement and equal probabilities, is the same as that considered in [1] and [2].
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Roy et al. (1960) studied this question.