An estimator is said to be consistent if for any ε > 0,
That is
is consistent if, as the sample size gets larger, it is less and less likely that
will be further than ε from the true value of θ.Show that
is a consistent estimator of μ when σ2 < ∞ by using Chebyshev’s inequality from Exercise 44 of Chapter 3. Hint: The inequality can be rewritten in the form
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