4.4
Fisher information
It is important to know how good our estimators are. We would
like our estimator to have as small variance as possible. There
is a useful lower bound on variances of the parameter estimators
called
Cram
è
r-Rao bound
. Let us first introduce the
Fisher information matrix
with the components defined by
The Cramèr-Rao bound states that for
unbiased
estimators the covariance matrix of the estimators
. (The inequality
for matrices means that the matrix
is nonnegative definite.)
A very important property of the ML estimators
is that asymptotically (i.e., for a signal-to-noise ratio tending
to infinity) they are (i) unbiased, and (ii) they have a Gaussian
distribution with covariance matrix equal to the inverse of the
Fisher information matrix.
4.4.1
Gaussian case
In the case of Gaussian noise the components
of the Fisher matrix are given by
For the case of the general gravitational-wave signal defined in
Equation (12) the set of the signal parameters
splits naturally into extrinsic and intrinsic parameters:
. Then the Fisher matrix can be written in terms of block
matrices for these two sets of parameters as
where the top left block corresponds to the extrinsic parameters,
the bottom right block corresponds to the intrinsic parameters,
the superscript T denotes here transposition over the extrinsic
parameter indices, and the dot stands for the matrix
multiplication with respect to these parameters. Matrix
is given by Equation (31), and the matrices
and
are defined as follows:
The covariance matrix
, which approximates the expected covariances of the ML parameter
estimators, is defined as
. Using the standard formula for the inverse of a block
matrix
[57
]
we have
where
We call
(the
Schur complement
of
) the
projected Fisher matrix
(onto the space of intrinsic parameters). Because the projected
Fisher matrix is the inverse of the intrinsic-parameter submatrix
of the covariance matrix
, it expresses the information available about the intrinsic
parameters that takes into account the correlations with the
extrinsic parameters. Note that
is still a function of the putative extrinsic parameters.
We next define the
normalized projected Fisher matrix
where
is the signal-to-noise ratio. From the Rayleigh principle
[57
]
follows that the minimum value of the component
is given by the smallest eigenvalue (taken with respect to the
extrinsic parameters) of the matrix
. Similarly, the maximum value of the component
is given by the largest eigenvalue of that matrix. Because the
trace of a matrix is equal to the sum of its eigenvalues, the
matrix
where the trace is taken over the extrinsic-parameter indices,
expresses the information available about the intrinsic
parameters, averaged over the possible values of the extrinsic
parameters. Note that the factor 1/4 is specific to the case of
four extrinsic parameters. We call
the
reduced Fisher matrix
. This matrix is a function of the intrinsic parameters alone. We
see that the reduced Fisher matrix plays a key role in the signal
processing theory that we review here. It is used in the
calculation of the threshold for statistically significant
detection and in the formula for the number of templates needed
to do a given search.
For the case of the signal
the normalized projected Fisher matrix
is independent of the extrinsic parameters
and
, and it is equal to the reduced matrix
[65
]
. The components of
are given by
where
is the Fisher matrix for the signal
.