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∙Because of examples like this – and others where we cannot computethe mean or the variance – we usually establish consistency using thelaw of large numbers and the algebra of plims.14
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EXAMPLE: Consider thePoissonexample where we want toestimatePX0exp−. The natural estimator iŝnexp−X̄n. Because 0≤̂n≤1,Ênexists, but we cannoteasily compute it. But it is easy to verify that̂nis consistent becausethe plim passes through continuous nonlinear functions (Slutsky’sTheorem):plim̂nplimexp−X̄nexp−plimX̄nexp−≡.∙If we have a sequence of vectorŝn:n1,2,...then consistencyis defined element by element.15