4 Maximum-likelihood Detection in Gaussian Noise

In this section, we study the detection of a deterministic gravitational-wave signal h(t;𝜃) of the general form given by Eq. (32View Equation) and the estimation of its parameters 𝜃 using the maximum-likelihood (ML) principle. We assume that the noise n(t) in the detector is a zero-mean, Gaussian, and stationary random process. The data x in the detector, in the case when the gravitational-wave signal h(t;𝜃) is present, is x (t;𝜃 ) = n (t) + h(t;𝜃). The parameters 𝜃 = (a,ξ) of the signal (32View Equation) split into extrinsic (or amplitude) parameters a and intrinsic ones ξ.

 4.1 The ℱ-statistic
  4.1.1 Targeted searches
 4.2 Signal-to-noise ratio and the Fisher matrix
 4.3 False alarm and detection probabilities
  4.3.1 False alarm and detection probabilities for known intrinsic parameters
  4.3.2 False alarm probability for unknown intrinsic parameters
  4.3.3 Detection probability for unknown intrinsic parameters
 4.4 Number of templates
  4.4.1 Covering problem
 4.5 Suboptimal filtering
 4.6 Algorithms to calculate the ℱ-statistic
  4.6.1 The two-step procedure
  4.6.2 Evaluation of the ℱ-statistic
 4.7 Accuracy of parameter estimation
  4.7.1 Fisher-matrix-based assessments
  4.7.2 Comparison with the Cramèr–Rao bound
 4.8 Upper limits

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