variance of any unbiased estimator and it expresses a fundamental limitation
on the accuracy with which a parameter can be estimated. We also note that
this lower bound is, in general, a function of , the true parameter value.
Several remarks in connection with the Crame ́r–Rao lower bound (CRLB)
are now in order.
Remark 1: the expectation in Equation (9.26) is equivalent to
,or
This alternate expression offers computational advantages in some cases.
Remark 2: the result given by Equation (9.26) can be extended easily to
multiple parameter cases. Let 1 , 2 ,..., and be the unknown
parameters in which are to be estimated on the basis of a
sample of size n. In vector notation, we can write
with corresponding vector unbiased estimator
F ollowing similar steps in the derivation of Equation (9.26), we can show that
the Crame ́r–Rao inequality for multiple parameters is of the form
where^1 is the inverse of matrix for which the elements are
Equation (9.39) implies that
where is the jjth element of^1.
Remark 3: the CRLB can be transformed easily under a transformation of
the parameter. Suppose that, instead of , parameter is of interest,
Parameter Estimation 269
.
Efq^2 lnfX;)/q^2 g
^2 ^nE
q^2 lnf
X;
q^2
1
: 9 : 36
.
mmn)
fx; 1 ,...,m),
qT 1 2 m;
9 : 37
Q^T^ 1 ^ 2 ^m:
9 : 38
covfQ^g
^1
n
; 9 : 39
ijE
qlnf
X;q
qi
qlnf
X;q
qj
; i;j 1 ; 2 ;...;m:
9 : 40
varf^jg
^1 jj
n
1
njj
; j 1 ; 2 ;...;m;
9 : 41
^1 )jj
g)
.