Because of examples like this and others where we

Info icon This preview shows pages 14–22. Sign up to view the full content.

Because of examples like this – and others where we cannot compute the mean or the variance – we usually establish consistency using the law of large numbers and the algebra of plims. 14
Image of page 14

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

EXAMPLE : Consider the Poisson example where we want to estimate P X 0 exp . The natural estimator is ̂ n exp X ̄ n . Because 0 ̂ n 1, E ̂ n exists, but we cannot easily compute it. But it is easy to verify that ̂ n is consistent because the plim passes through continuous nonlinear functions (Slutsky’s Theorem): plim ̂ n plim exp X ̄ n  exp plim X ̄ n  exp . If we have a sequence of vectors ̂ n : n 1,2,... then consistency is defined element by element. 15
Image of page 15