Page 367 - Probability and Statistical Inference

344 7. Point Estimation

-1
2
2
estimator of σ coincides with (n - 1)n S . Note, however, that is
sufficient for θθ θθ θ too. !
Example 7.2.3 Suppose that X , ..., X are iid N(0, σ ) where θ = σ is
2
2
n
1
χ
unknown, 0 < σ < ∞. Here = ℜ and Θ = ℜ . Observe that η ≡ η (θ) =
+
+
1
1
E [X ] = 0 for all θθ θθ θ. In this situation, it is clear that the equation given
θ
1
by the first moment in (7.2.1) does not lead to anything interesting and
one may arbitrarily move to use the second moment. Note that
so that (7.2.1) will now lead to
After such ad hoc adjustment, the method of moment estimator turns out to
2
be sufficient for σ . !
The method of moments is an ad hoc way to find estimators. Also,
this method may not lead to estimators which are functions of
minimal sufficient statistics for θ. Look at Examples 7.2.4-7.2.5.
If any of the theoretical moments η , η , ..., η is zero, then one will continue
1 2 k
to work with the first k non-zero theoretical moments so that (7.2.1) may
lead to sensible solutions. Of course, there is a lot of arbitrariness in this
approach.
Example 7.2.4 Suppose that X , ..., X are iid Poisson(λ) where θ = λ is
1
n
unknown, 0 < λ < ∞. Here χ = {0, 1, 2, ...} and Θ = ℜ . Now, η ≡ η (θ) =
+
1 1
E [X ] = λ and Suppose that instead of
θ 1
starting with η in (7.2.1), we start with η and equate this with
2
1
This then provides the equation
which leads to the estimator However, if
we had started with η , we would have ended up with the estimator , a
1
minimal sufficient statistic for η. The first estimator is not sufficient for η.
From this example, one can feel the sense of arbitrariness built within this
methodology. !
Example 7.2.5 Suppose that X , ..., X are iid Uniform (0, θ) where θ(> 0)
1
n
is the unknown parameter. Here η ≡ η (θ) = E [X ] = ½θ so that by equating
1
1
1
?
this with we obtain which is not sufficient for ?. Recall that X is a minimal
n:n
sufficient statistic for θ. !
7.2.2 The Method of Maximum Likelihood
The method of moments appeared quite simple but it was ad hoc and arbi-
trary in its approach. In the Example 7.2.3 we saw that we could not equate