Fisher information matrix
If there is only one parameter involved, then is simply called the Fisher information or information of .
If belongs to the exponential family, . Furthermore, with some regularity conditions imposed, we have
Now, in linear regression model with constant variance , it can be shown that the Fisher information matrix is
where X is the design matrix of the regression model.
In general, the Fisher information meansures how much “information” is known about a parameter . If is an unbiased estimator of , it can be shown that
This is known as the Cramer-Rao inequality, and the number is known as the Cramer-Rao lower bound. The smaller the variance of the estimate of , the more information we have on . If there is more than one parameter, the above can be generalized by saying that
is positive semidefinite, where is the Fisher information matrix.
|Title||Fisher information matrix|
|Date of creation||2013-03-22 14:30:15|
|Last modified on||2013-03-22 14:30:15|
|Last modified by||CWoo (3771)|
|Defines||Cramer-Rao lower bound|