### 4.1 The -statistic

For the gravitational-wave signal of the form given in Eq. (32) the log likelihood function (46) can be written as
where the components of the column matrix and the square matrix are given by
The ML equations for the extrinsic parameters , , can be solved explicitly to show that the ML estimators of the parameters are given by
Replacing the extrinsic parameters in Eq. (59) by their ML estimators , we obtain the reduced log likelihood function,
that we call the -statistic. The -statistic depends nonlinearly on the intrinsic parameters of the signal, it does not depend on the extrinsic parameters .

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.

#### 4.1.1 Targeted searches

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].