Analytic expressions for the Cramér-Rao bound are well known. The developments appear in texts such as Van Trees (1968), Whalen (1971), Kendall and Stuart (1973) and Snyder (1975). For the one-parameter problem where a is to be estimated by an unbiased estimate, â, the simplest possible case, the CR bound is
where
and E{}denotes a statistical average over what is random in d. The left side of (1) is the mean square error for the estimate and the right side is the bound. No estimate for a can have a mean square error smaller than the bound.
From (1) we observe the following regarding changes in a: if the PDF change is large the error will be small; if, however, the PDF change is small the error is large. The CR bound estimates the sensitivity of the data to changes in a.
The multi-parameter, biased estimate requires calculation of the Fisher information matrix, F, a bias vector, b, and a correlation matrix C. They are defined by the elements
Then the CR bound on is the
diagonal of the matrix
R:
The right hand matrix
is commonly referred to as the ``variance-covariance matrix'' (Bury 1976).