∙
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.
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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.
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