We first present the false alarm and detection pdfs when the intrinsic parameters of the signal are known. In this case the statistic is a quadratic form of the random variables that are correlations of the data. As we assume that the noise in the data is Gaussian and the correlations are linear functions of the data, is a quadratic form of the Gaussian random variables. Consequently -statistic has a distribution related to the distribution. One can show (see Section III B in ) that for the signal given by Equation (14), has a distribution with 4 degrees of freedom when the signal is absent and noncentral distribution with 4 degrees of freedom and non-centrality parameter equal to signal-to-noise ratio when the signal is present.
As a result the pdfs and of when the intrinsic parameters are known and when respectively the signal is absent and present are given by[94, 44], . We see that when the noise in the detector is Gaussian and the intrinsic parameters are known, the probability of detection of the signal depends on a single quantity: the optimal signal-to-noise ratio .
Next we return to the case when the intrinsic parameters are not known. Then the statistic given by Equation (35) is a certain generalized multiparameter random process called the random field (see Adler’s monograph  for a comprehensive discussion of random fields). If the vector has one component the random field is simply a random process. For random fields we can define the autocovariance function just in the same way as we define such a function for a random process:
One can estimate the false alarm probability in the following way . The autocovariance function tends to zero as the displacement increases (it is maximal for ). Thus we can divide the parameter space into elementary cells such that in each cell the autocovariance function is appreciably different from zero. The realizations of the random field within a cell will be correlated (dependent), whereas realizations of the random field within each cell and outside the cell are almost uncorrelated (independent). Thus the number of cells covering the parameter space gives an estimate of the number of independent realizations of the random field. The correlation hypersurface is a closed surface defined by the requirement that at the boundary of the hypersurface the correlation equals half of its maximum value. The elementary cell is defined by the equation, Appendix B) that the matrix is precisely equal to the reduced Fisher matrix given by Equation (43).
Let be the number of the intrinsic parameters. If the components of the matrix are constant (independent of the values of the parameters of the signal) the above equation is an equation for a hyperellipse. The -dimensional Euclidean volume of the elementary cell defined by Equation (57) equalslinear function of the intrinsic parameters .
To estimate the number of cells in the case when the components of the matrix are not constant, i.e. when they depend on the values of the parameters, we write Equation (59) as, where an analogous integral formula was proposed for the number of templates needed to perform a search for gravitational-wave signals from coalescing binaries.
The concept of number of cells was introduced in  and it is a generalization of the idea of an effective number of samples introduced in  for the case of a coalescing binary signal.
We approximate the probability distribution of in each cell by the probability when the parameters are known [in our case by probability given by Equation (46)]. The values of the statistic in each cell can be considered as independent random variables. The probability that does not exceed the threshold in a given cell is , where is given by Equation (48). Consequently the probability that does not exceed the threshold in all the cells is . The probability that exceeds in one or more cell is thus given by.
It was shown (see ) that for any finite and , Equation (61) provides an upper bound for the false alarm probability. Also in  a tighter upper bound for the false alarm probability was derived by modifying a formula obtained by Mohanty . The formula amounts essentially to introducing a suitable coefficient multiplying the number of cells .
When the signal is present a precise calculation of the pdf of is very difficult because the presence of the signal makes the data random process non-stationary. As a first approximation we can estimate the probability of detection of the signal when the parameters are unknown by the probability of detection when the parameters of the signal are known [given by Equation (49)]. This approximation assumes that when the signal is present the true values of the phase parameters fall within the cell where has a maximum. This approximation will be the better the higher the signal-to-noise ratio is.
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License.