Page 286 - Fundamentals of Probability and Statistics for Engineers

Parameter Estimation 269

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
2
2
Efq ln f X; )/q g , or
2
q ln f X; † 1
2
nE : 9:36†
^ 2
q
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 m m n) be the unknown
parameters in f x; 1 , ... , m ), which are to be estimated on the basis of a
sample of size n. In vector notation, we can write

T
q ˆ‰ 1 2 m Š; 9:37†
with corresponding vector unbiased estimator

^ T ^ ^ ^
Q ˆ‰ 1 2 m Š: 9:38†
Following 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

^
covfQg 1 ; 9:39†
n
where 1 is the inverse of matrix for which the elements are
q ln f X; q† q ln f X; q†

ij ˆ E ; i; j ˆ 1; 2; ... ; m: 9:40†
q i q j
Equation (9.39) implies that

1
† 1
^
varf j g jj ; j ˆ 1; 2; ... ; m; 9:41†
n n jj
1
where ) is the jjth element of 1 .
jj
. Remark 3: the CRLB can be transformed easily under a transformation of
the parameter. Suppose that, instead of , parameter ˆ g ) is of interest,

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