Page 360 - Fundamentals of Probability and Statistics for Engineers
P. 360
Linear Models and Linear Regression 343
2 2
I, I being
then E is a zero-mean random vector with covariance matrix L ,
the n n identity matrix.
The mean and variance of estimator Q are now easily determined. In view of
^
Equations (11.17) and (11.19), we have
1
^
T
T
EfQg
C C C EfYg
1
T
T
C C C Cq EfEg
11:20
1
T
T
C C
C Cq q:
^
^
Hence, estimators A and B for and , respectively, are unbiased.
The covariance matrix associated with Q is given by, as seen from Equation
^
(11.17),
^
^
T
^
covfQg Ef
Q q
Q q g
1
1
T
T
T
C C C covfYgC
C C :
But cov Y g 2 I; we thus have
f
^
1
1
2
T
2
T
T
T
covfQg
C C C C
C C 1
C C :
11:21
The diagonal elements of the matrix in Equation (11.21) give the variances of
^
A ^ and B . In terms of the elements of C, we can write
1
n X
n
" #" #
X
^
varfAg 2 x 2 n
x i x 2 ;
11:22
i
i1 i1
1
" #
n
X
^
varfBg 2
x i x 2 :
11:23
i1
It is seen that these variances decrease as sample size n increases, according to 1/n.
Thus, it followsfrom our discussion in Chapter 9that theseestimatorsareconsistent –
a desirable property. We further note that, for a fixed n, the variance of B ^ can be
reduced by selecting the x i in such a way that the denominator of Equation (11.23) is
maximized; this can be accomplished by spreading the x i as far apart as possible. In
Example 11.1, for example, assuming that we are free to choose the values of x i , the
quality of ^ is improved if one-half of the x readings are taken at one extreme of the
temperature range and the other half at the other extreme. However, the sampling
^
A
strategy for minimizing var( ) for a fixed n is to make x as close to zero as possible.
Are the variances given by Equations (11.22) and (11.23) minimum variances
associated with any unbiased estimators for and ? An answer to this import-
ant question can be found by comparing the results given by Equations (11.22)
TLFeBOOK
