Some Examples of Minimum Variance Unbiased Estimates

Tenenbein (1971) presented an interesting example in which he constructed a minimum variance unbiased estimate of a certain integer valued parameter. His example is reviewed in the light of the fact that his "estimate" takes on non-integer values. Below, in such a case, the term "estimate" is reserved for integervalued functions of the observations. With this reservation, it will be shown that no unbiased "estimates" exist in Tenenbein's example. An alternative example is suggested which retains the pedagogical value of his example without the accompanying faults. Finally, an example is given in which a unique minimum variance unbiased estimate exists despite lack of coinpleteness of the family of distributions of a minimal sufficient statistic.