The procedure to detect the gravitational-wave signal of the form (32) and estimate its parameters
consists of two parts. The first part is to find the (local) maxima of the
-statistic (62
) in the intrinsic
parameters space. The ML estimators
of the intrinsic parameters
are those values of
for which
the
-statistic attains a maximum. The second part is to calculate the estimators
of
the extrinsic parameters
from the analytic formula (61
), where the matrix
and the
correlations
are calculated for the parameters
replaced by their ML estimators
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
-statistic.
The -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
[see Eq. (30
)]. To detect the signal one calculates the
-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 (61
).
In [109] it was shown that the maximum-likelihood -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
-statistic) was obtained. Monte Carlo simulations
for signals with random (isotropic) spin-axis orientations show that the
-statistic is more
powerful (in terms of its expected detection probability) than the
-statistic. A modified
version of the
-statistic that can be more powerful than the original one has been studied
in [20].
http://www.livingreviews.org/lrr-2012-4 |
Living Rev. Relativity 15, (2012), 4
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