4.7 Accuracy of parameter estimation

4.7.1 Fisher-matrix-based assessments

Fisher matrix has been extensively used to assess the accuracy of estimation of astrophysically-interesting parameters of different gravitational-wave signals. For ground-based interferometric detectors, the first calculations of the Fisher matrix concerned gravitational-wave signals from inspiralling compact binaries (made of neutron stars or black holes) in the leading-order quadrupole approximation [50, 76, 63Jump To The Next Citation Point] and from quasi-normal modes of Kerr black hole [48].

Cutler and Flanagan [41Jump To The Next Citation Point] initiated the study of the implications of the higher-order post-Newtonian (PN) phasing formula as applied to the parameter estimation of inspiralling binary signals. They used the 1.5PN phasing formula to investigate the problem of parameter estimation, both for spinning and non-spinning binaries, and examined the effect of the spin-orbit coupling on the estimation of parameters. The effect of the 2PN phasing formula was analyzed independently by Poisson and Will [106Jump To The Next Citation Point] and Królak, Kokkotas and Schäfer [75Jump To The Next Citation Point]. In both cases the focus was to understand the leading-order spin-spin coupling term appearing at the 2PN level when the spins were aligned perpendicularly to the orbital plane. Compared to [75], [106] also included a priori information about the magnitude of the spin parameters, which then leads to a reduction in the rms errors in the estimation of mass parameters. The case of a 3.5PN phasing formula was studied in detail by Arun et al. [17]. Inclusion of 3.5PN effects leads to an improved estimate of the binary parameters. Improvements are relatively smaller for lighter binaries. More recently the Fisher matrix was employed to assess the errors in estimating the parameters of nonspinning black-hole binaries using the complete inspiral-merger-ring-down waveforms [7].

Various authors have investigated the accuracy with which the LISA detector can determine binary parameters including spin effects. Cutler [40] determined LISA’s angular resolution and evaluated the errors of the binary masses and distance considering spins aligned or anti-aligned with the orbital angular momentum. Hughes [60] investigated the accuracy with which the redshift can be estimated (if the cosmological parameters are derived independently), and considered the black-hole ring-down phase in addition to the inspiralling signal. Seto [128] included the effect of finite armlength (going beyond the long wavelength approximation) and found that the accuracy of the distance determination and angular resolution improve. This happens because the response of the instrument when the armlength is finite depends strongly on the location of the source, which is tightly correlated with the distance and the direction of the orbital angular momentum. Vecchio [140] provided the first estimate of parameters for precessing binaries when only one of the two supermassive black holes carries spin. He showed that modulational effects decorrelate the binary parameters to some extent, resulting in a better estimation of the parameters compared to the case when spins are aligned or antialigned with orbital angular momentum. Hughes and Menou [61] studied a class of binaries, which they called “golden binaries,” for which the inspiral and ring-down phases could be observed with good enough precision to carry out valuable tests of strong-field gravity. Berti, Buonanno and Will [29] have shown that inclusion of non-precessing spin-orbit and spin-spin terms in the gravitational-wave phasing generally reduces the accuracy with which the parameters of the binary can be estimated. This is not surprising, since the parameters are highly correlated, and adding parameters effectively dilutes the available information.

Extensive study of accuracy of parameter estimation for continuous gravitational-wave signals from spinning neutron stars was performed in [64]. In [129] Seto used the Fisher matrix to study the possibility of determining distances to rapidly rotating isolated neutron stars by measuring the curvature of the wave fronts.

4.7.2 Comparison with the Cramèr–Rao bound

In order to test the performance of the maximization method of the ℱ-statistic it is useful to perform Monte Carlo simulations of the parameter estimation and compare the simulated variances of the estimators with the variances calculated from the Fisher matrix. Such simulations were performed for various gravitational-wave signals [73, 26, 65Jump To The Next Citation Point, 36]. In these simulations one observes that, above a certain signal-to-noise ratio, called the threshold signal-to-noise ratio, the results of the Monte Carlo simulations agree very well with the calculations of the rms errors from the inverse of the Fisher matrix. However, below the threshold signal-to-noise ratio they differ by a large factor. This threshold effect is well known in signal processing [139]. There exist more refined theoretical bounds on the rms errors that explain this effect, and they were studied in the context of the gravitational-wave signals from coalescing binaries [98Jump To The Next Citation Point].

Use of the Fisher matrix in the assessment of accuracy of the parameter estimation has been critically examined in [138], where a criterion has been established for the signal-to-noise ratio above which the inverse of the Fisher matrix approximates well covariances of the parameter estimators. In [148, 142Jump To The Next Citation Point] the errors of ML estimators of parameters of gravitational-wave signals from nonspinning black-hole binaries were calculated analytically using a power expansion of the bias and the covariance matrix in inverse powers of the signal-to-noise ratio. The first-order term in this covariance matrix expansion is the inverse of the Fisher information matrix. The use of higher-order derivatives of the likelihood function in these expansions makes the errors prediction sensitive to the secondary lobes of the pdf of the ML estimators. Conditions for the validity of the Cramèr–Rao lower bound are discussed in [142] as well, and some new features in regions of the parameter space so far not explored are predicted (e.g., that the bias can become the most important contributor to the parameters errors for high-mass systems with masses 200M ⊙ and above).

There exists a simple model that explains the deviations from the covariance matrix and reproduces well the results of the Monte Carlo simulations (see also [25]). The model makes use of the concept of the elementary cell of the parameter space that we introduced in Section 4.3.2. The calculation given below is a generalization of the calculation of the rms error for the case of a monochromatic signal given by Rife and Boorstyn [116].

When the values of parameters of the template that correspond to the maximum of the functional ℱ fall within the cell in the parameter space where the signal is present, the rms error is satisfactorily approximated by the inverse of the Fisher matrix. However, sometimes, as a result of noise, the global maximum is in the cell where there is no signal. We then say that an outlier has occurred. In the simplest case we can assume that the probability density of the values of the outliers is uniform over the search interval of a parameter, and then the rms error is given by

2 Δ2- σout = 12, (123 )
where Δ is the length of the search interval for a given parameter. The probability that an outlier occurs will be higher the lower the signal-to-noise ratio is. Let q be the probability that an outlier occurs. Then the total variance σ2 of the estimator of a parameter is the weighted sum of the two errors
2 2 2 σ = σoutq + σCR(1 − q), (124 )
where σCR is the rms errors calculated from the covariance matrix for a given parameter. One can show [65Jump To The Next Citation Point] that the probability q can be approximated by the following formula:
∫ ∞ (∫ ℱ )Ncells− 1 q = 1 − p1(ρ,ℱ ) p0(y)dy dℱ , (125 ) 0 0
where p0 and p1 are the pdfs of the ℱ-statistic (for known intrinsic parameters) when the signal is absent or present in data, respectively [they are given by Eqs. (84View Equation) and (85View Equation)], and where N cells is the number of cells in the intrinsic parameter space. Eq. (125View Equation) is in good but not perfect agreement with the rms errors obtained from the Monte Carlo simulations (see [65]). There are clearly other reasons for deviations from the Cramèr–Rao bound as well. One important effect (see [98]) is that the functional ℱ has many local subsidiary maxima close to the global one. Thus, for a low signal-to-noise ratio the noise may promote the subsidiary maximum to a global one.
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