Fundamentals of Probability and Statistics for Engineers

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



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