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380 7. Point Estimation

7.7 Consistent Estimators

The consistency is a large sample property of an estimator. R. A. Fisher had
introduced the concept of consistency in his 1922 paper. Let us suppose that
we have {T = T (X , ..., X ); n ≥ 1}, a sequence of estimators for some
n n 1 n
unknown real valued parametric function T(θθ θθ θ) where θθ θθ θ ∈ Θ ⊆ ℜ . We empha-
k
size the dependence of the estimators on the sample size n by indexing with
subscript n. It may help to recall the concept of the convergence in probabil-
ity, denoted by which was laid out by the Definition 5.2.1.
Definition 7.7.1 Consider {T ≡ T (X , ..., X ); n ≥ 1}, a sequence of
n
n
n
1
estimators for some unknown real valued parametric function T(θθ θθ θ) where θθ θθ θ ∈
Θ ⊆ ℜ . Then, T is said to be consistent for T(θθ θθ θ) if and only if as
k
n
n → ∞ . Also, T is called inconsistent for T(θθ θθ θ) if T is not consistent for T(θθ θθ θ).
n n
An estimator T may be biased for T(θθ θθ θ) and yet T n
n
may be consistent for T(θθ θθ θ).
Estimators found by the method of moments are often smooth functions
of averages of powers of the X’s and so they are consistent for the associated
parametric functions. One should verify that all the estimators derived in the
Examples 7.2.1-7.2.5 are indeed consistent for the parameters of interest.
The MLE’s derived in the Examples 7.2.6-7.2.7 and also in the Examples
7.2.9-7.2.12 are all consistent.
Let us particularly draw attention to the Examples 7.2.5 and 7.2.9. In the
Uniform(0, θ) case, the method of moment estimator (Example 7.2.5) for θ
turned out to be and by the Weak WLLN (Theorem 5.2.1) it follows
that as n → ∞. That is, T is a consistent estimator of θ even though
n
T is not exclusively a function of the minimal sufficient statistic X . The
n n:n
MLE (Example 7.2.9) of θ is X and recall from Example 5.2.5
n:n
that as n → ∞. Thus, X is a consistent estimator for θ. Note that X is not
n:n
n:n
unbiased for θ.
All the UMVUE’s derived in Section 7.5 are consistent for the parameters
of interest. Their verifications are left as exercises. Let us, however, look at
the Example 7.5.4 where the unbiased estimator was
-λ
found for the parametric function e in the case of the Poisson(λ) distribu-
tion. Let us rewrite

where V = (1-n ) → e as n → ∞. In this deliberation,
-1
-1 n
n