Non-Existence of Consistent Estimator Sequences and Unbiased Estimates: A Practical Example

It is often thought that identifiability implies existence of consistent estimator sequences. A rather artificial counter example is given in [7]. We here consider a case which often arises in experimental and survey practice. The example concerns a model with intraclass correlation ρ. For ρ negative an indefinitely large sequence of observations cannot arise from such a model and so the discussion of consistency is restricted to ρ > 0. For any non-degenerate range of ρ we show that no unbiased estimate exists for the variance of the mean of the observations. Certain other aspects of estimation in models of this sort are considered in [4].