4.1 The ℱ-statistic

For the gravitational-wave signal h(t;a,ξ) of the form given in Eq. (32View Equation) the log likelihood function (46View Equation) can be written as
T 1 T log Λ[x;a,ξ] = a ⋅ N[x;ξ ] −-a ⋅ M (ξ ) ⋅ a, (59 ) 2
where the components of the column n × 1 matrix N and the square n × n matrix M are given by
Nk[x;ξ] := (x|hk (t;ξ )), Mkl(ξ) := (hk(t;ξ)|hl(t;ξ)), k, l = 1, ...,n. (60 )
The ML equations for the extrinsic parameters a, ∂ logΛ [x;a,ξ]∕∂a = 0, can be solved explicitly to show that the ML estimators ˆa of the parameters a are given by
−1 ˆa[x; ξ] = M (ξ ) ⋅ N [x;ξ]. (61 )
Replacing the extrinsic parameters a in Eq. (59View Equation) by their ML estimators ˆa, we obtain the reduced log likelihood function,
[ ] 1 T − 1 ℱ [x;ξ] := log Λ x; ˆa[x; ξ],ξ = -N [x;ξ] ⋅ M(ξ ) ⋅ N [x; ξ], (62 ) 2
that we call the ℱ-statistic. The ℱ-statistic depends nonlinearly on the intrinsic parameters ξ of the signal, it does not depend on the extrinsic parameters a.

The procedure to detect the gravitational-wave signal of the form (32View Equation) and estimate its parameters consists of two parts. The first part is to find the (local) maxima of the ℱ-statistic (62View Equation) in the intrinsic parameters space. The ML estimators ˆ ξ of the intrinsic parameters ξ are those values of ξ for which the ℱ-statistic attains a maximum. The second part is to calculate the estimators ˆa of the extrinsic parameters a from the analytic formula (61View Equation), where the matrix M and the correlations N are calculated for the parameters ξ replaced by their ML estimators ˆξ obtained from the first part of the analysis. We call this procedure the maximum likelihood detection. See Section 4.6 for a discussion of the algorithms to find the (local) maxima of the ℱ-statistic.

4.1.1 Targeted searches

The ℱ-statistic can also be used in the case when the intrinsic parameters are known. An example of such an analysis called a targeted search is the search for a gravitational-wave signal from a known pulsar. In this case assuming that gravitational-wave emission follows the radio timing, the phase of the signal is known from pulsar observations and the only unknown parameters of the signal are the amplitude (or extrinsic) parameters a [see Eq. (30View Equation)]. To detect the signal one calculates the ℱ-statistic for the known values of the intrinsic parameters and compares it to a threshold [67]. When a statistically-significant signal is detected, one then estimates the amplitude parameters from the analytic formulae (61View Equation).

In [109] it was shown that the maximum-likelihood ℱ-statistic can be interpreted as a Bayes factor with a simple, but unphysical, amplitude prior (and an additional unphysical sky-position weighting). Using a more physical prior based on an isotropic probability distribution for the unknown spin-axis orientation of emitting systems, a new detection statistic (called the ℬ-statistic) was obtained. Monte Carlo simulations for signals with random (isotropic) spin-axis orientations show that the ℬ-statistic is more powerful (in terms of its expected detection probability) than the ℱ-statistic. A modified version of the ℱ-statistic that can be more powerful than the original one has been studied in [20].


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