Page 76 - Computational Statistics Handbook with MATLAB

62 Computational Statistics Handbook with MATLAB

[
2
2
MSE T() = ET –([ 2 2Tθ + θ )] = ET ] – 2θET[] + θ 2 . (3.16)
By adding and subtracting ET[]( ) 2 to the right hand side of Equation 3.16,
we have the following

[
2
2
2
MSE T() = ET ] – ( ET[]) + ( ET[]) – 2θET[] + θ 2 . (3.17)
The first two terms of Equation 3.17 are the variance of T, and the last three
terms equal the squared bias of our estimator. Thus, we can write the mean
squared error as

2
[
2
MSE T() = ET ] – ( ET[]) + ( ET[] – θ) 2
(3.18)
2
= VT() + [ bias T()] .
Since the mean squared error is based on the variance and the squared bias,
the error will be small when the variance and the bias are both small. When
T is unbiased, then the mean squared error is equal to the variance only. The
concepts of bias and variance are important for assessing the performance of
any estimator.

e
RReell laat aatt ti ii ivveEfficiencyeEfficiency
Rel
Re
vveEfficiencyEfficiency
Another measure we can use to compare estimators is called efficiency, which
is defined using the MSE. For example, suppose we have two estimators
,
(
,
,
(
,
T 1 = t 1 X 1 … X n ) and T 2 = t 2 X 1 … X n ) for the same parameter. If the
MSE of one estimator is less than the other (e.g., MSE T 1 ) < MSE T 2 ) ), then
(
(
.
T 1 is said to be more efficient than T 2
is given by
The relative efficiency of T 1 to T 2
MSE T 2 )
(
(
,
eff T 1 T 2 ) = ----------------------- . (3.19)
MSE T 1 )
(
is a more efficient estimator of the
If this ratio is greater than one, then T 1
parameter.
SStandardErEr
SErtandardErr
roor
Standard tandard rr oorr r
We can get a measure of the precision of our estimator by calculating the stan-
dard error. The standard error of an estimator (or a statistic) is defined as the
standard deviation of its sampling distribution:
SE T() = VT() = σ T .
© 2002 by Chapman & Hall/CRC