Page 375 - Probability and Statistical Inference

352 7. Point Estimation

defined as follows:

Based on X , ..., X , one can certainly form many other rival estimators for µ.
1
4
Observe that E (T ) = 2µ, E (T ) = µ, E (T ) = µ, E (T ) = 2/3µ, E (T ) = µ
µ
3
µ
2
1
µ
µ
µ
5
4
and E (T ) = µ. Thus, T and T are both biased estimators of µ, but T , T , T 5
1
µ
6
4
3
2
and T are unbiased estimators of µ. If we wish to estimate µ unbiasedly, then
6
among T through T , we should only include T , T , T , T for further consid-
6
3
5
6
2
1
erations.
Definition 7.3.3 Suppose that the real valued statistic T ≡ T(X , ..., X ) is
n
1
an estimator of T(θ). Then, the mean squared error (MSE) of the estimator T,
2
sometimes denoted by MSE , is given by E θ [(T - T(θ)) ]. If T is an unbiased
T
estimator of T(θ), then the MSE is the variance of T, denoted by V (T).
θ
Note that we have independence between the X’s. Thus, utilizing the Theo-
rem 3.4.3 in the case of the example we worked with little earlier, we obtain
In other words, these are the mean squared errors associated with the unbi-
ased estimators T , T , T and T .
3
6
2
5
How would one evaluate the MSE of the biased estimators T and T ? The
4
1
following result will help in this regard.
Theorem 7.3.1 If a statistic T is used to estimate a real valued parametric
function T(θ), then MSE , the MSE associated with T, is given by
T
which amounts to saying that the MSE is same as the variance plus the square
of the bias.