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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 G with the components defined by
[@ ln /\ @ ln /\ ] [@2 ln /\ ] Gij := E ------------ = - E ------- . (34) @hi @hj @hi@hj
The Cramèr-Rao bound states that for unbiased estimators the covariance matrix of the estimators - 1 C > G . (The inequality A > B for matrices means that the matrix A - B 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

( | ) @h |@h Gij = @h-||@h-- . (35) i j
For the case of the general gravitational-wave signal defined in Equation (12View Equation) the set of the signal parameters h splits naturally into extrinsic and intrinsic parameters: h = (a(k),qm) . Then the Fisher matrix can be written in terms of block matrices for these two sets of parameters as
( M F .a ) G = T T T , (36) a .F a .S .a
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 M is given by Equation (31View Equation), and the matrices F and S are defined as follows:
( | ) ( | ) (k)(l) (k)||@h(l) (k)(l) @h(k)||@h(l) Fm := h | @qm , Smn := @qm |@qn . (37)

The covariance matrix C, which approximates the expected covariances of the ML parameter estimators, is defined as G -1 . Using the standard formula for the inverse of a block matrix  [57Jump To The Next Citation Point] we have

( M -1 + M -1 .(F .a) .G- 1 .(F .a)T .M -1 -M - 1 .(F .a) .G-1 ) C = , (38) - G-1 .(F .a)T .M -1 G -1
where
G := aT .(S - FT .M - 1 .F) .a. (39)
We call mn G (the Schur complement of M) 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 C, it expresses the information available about the intrinsic parameters that takes into account the correlations with the extrinsic parameters. Note that Gmn is still a function of the putative extrinsic parameters.

We next define the normalized projected Fisher matrix

T T -1 G := G--= a--.(S---F--.M----.F)-.a-, (40) n r2 aT .M .a
where V~ -T-------- r = a .M .a is the signal-to-noise ratio. From the Rayleigh principle  [57Jump To The Next Citation Point] follows that the minimum value of the component Gmn n is given by the smallest eigenvalue (taken with respect to the extrinsic parameters) of the matrix ((S - FT .M - 1 .F) .M -1)mn . Similarly, the maximum value of the component mn Gn 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
~ 1- [( T -1 ) - 1] G := 4 tr S - F .M .F .M , (41)
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 ~Gmn 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

m m m h(t;A0, f0,q ) = A0 g(t;q )cos (f(t;q )- f0) , (42)
the normalized projected Fisher matrix Gn is independent of the extrinsic parameters A0 and f0, and it is equal to the reduced matrix ~G   [65Jump To The Next Citation Point] . The components of ~G are given by
Gf0mGf0n ~Gmn = Gmn0 - -0-ff0--, (43) G 000
where Gij 0 is the Fisher matrix for the signal g(t;qm) cos(f(t;qm)- f ) 0 .
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