Consistent Estimator. Another way to measure the closeness of an estimator Θ to the parameter θ is in terms of consistency. If Θ_nis an estimator of θ based on a random sample of n observations, Θ_nis consistent for θ if

Thus, consistency is a large-sample property describing the limiting behavior of Θ_n as n tends to infinity. It is usually difficult to prove consistency using this definition, although it can be done from other approaches. To illustrate, show that X is a consistent estimator of μ (when σ² <ˆž) by using Chebyshev€™s inequality from the supplemental material on the Web.

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