Page 376 - Probability and Statistical Inference

7. Point Estimation 353

Proof Let us write E (T) = ξ(θ). Then, we have
θ

Now, since ξ(θ) - T(θ) is a fixed real number, we can write E [{T - ξ(θ)}{ξ(θ)
θ
- T(θ)}] = {ξ(θ) - T(θ)} E [{T - θ(θ)}] = 0. Hence the result follows. !
θ
Now, we can evaluate the MSE as V (T ) + [E (T ) - µ] = 2σ + (2µ -
2
2
T 1 µ 1 µ 1
µ) = µ + 2σ . The evaluation of MSE is left as the Exercise 7.3.6.
2
2
2
T 4
It is possible sometimes to have T and T which are respectively
1
2
biased and unbiased estimators of T(θ), but MSE < MSE . In
T 1 T 4
other words, intuitively a biased estimator may be preferable if its
average squared error is smaller. Look at the Example 7.3.1.
Example 7.3.1 Let X , ..., X be iid N(µ, σ ) where µ, σ are both un-
2
2
n
1
χ
known, θ = (µ, σ ), −∞ < µ < ∞, 0 < σ < ∞, n ≥ 2. Here = ℜ and Θ = ℜ ×
2
2
ℜ . Our goal is the estimation of a parametric function T(θ) = σ , the popula-
+
tion variance S = Consider the customary sample
2
2
2
variance We know that S is unbiased for σ . One will also recall that (n - 1)S /
2
σ is distributed as and hence V (S ) = 2σ (n - 1) . Next, consider an-
2
2
-1
4
θ
2
other estimator for σ , namely T = (n + which can be
2
2
rewritten as (n - 1)(n + 1) S . Thus, E (T) = (n - 1)(n + 1) σ ≠ σ and so
-1
2
-1
θ
T is a biased estimator of σ . Next, we evaluate
2
Then we apply the Theorem 7.3.1 to express MSE as
T
which is smaller than V (S ). That is, S is unbiased for σ and T is biased for
2
2
2
θ
σ but MSE is smaller than MSE 2 for all θ. Refer to the Exercise 7.3.2 to see
2
T s
2
how one comes up with an estimator such as T for σ . !
In the context of the example we had been discussing earlier in this
section, suppose that we consider two other estimators T = ½X and
7 1