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tion is extended by removing the restriction of unbiasedness, with the corresponding modification of the definition of admissibility: Now some other estimate is shown to remain admissible for all sampling designs. The result appears to have implications concerning the basic logic of sampling with varying probabilities. These however are not discussed here. 2. Notation. The notation used here is the same as that formulated in the Section 2 of the Part I of this paper and is not restated here. The definitions and preliminaries, as given in that section, also apply in the following discussion. In addition for convenience of discussion, here we assume that the units u of the population U are numbered, that is U = (Ui, . , UN), N being the total number of units u in U. As a result a sample s (Definition 2.2, Part I) can now be specified by the set of integers namely the serial numbers of the units u - s. Thus for Ur I s now we write r - s. Further, the variate value X(Ur) associated

Admissibility and Bayes Estimation in Sampling Finite Populations--V