Parameter Estimation

4 Parameter Estimation

Very often we know the waveform of the signal that we are searching for in the data in terms of a finite number of unknown parameters. We would like to find optimal procedures of estimating these parameters. An estimator of a parameter $θ$ is a function $ˆθ(x)$ that assigns to each data the “best” guess of the true value of $θ$ . Note that because $ˆθ(x)$ depends on the random data it is a random variable. Ideally we would like our estimator to be (i) unbiased, i.e., its expectation value to be equal to the true value of the parameter, and (ii) of minimum variance. Such estimators are rare and in general difficult to find. As in the signal detection there are several approaches to the parameter estimation problem. The subject is exposed in detail in reference [63 ]. See also [103 ] for a concise account.

4.1 Bayesian estimation
4.2 Maximum a posteriori probability estimation
4.3 Maximum likelihood estimation
  4.3.1 Gaussian case
4.4 Fisher information
  4.4.1 Gaussian case
4.5 False alarm and detection probabilities – Gaussian case
  4.5.1 Statistical properties of the $ℱ$ -statistic
  4.5.2 False alarm probability
  4.5.3 Detection probability
4.6 Number of templates
4.7 Suboptimal filtering
4.8 Algorithms to calculate the $ℱ$ -statistic
  4.8.1 The two-step procedure
  4.8.2 Evaluation of the $ℱ$ -statistic
  4.8.3 Comparison with the Cramèr–Rao bound
4.9 Upper limits
4.10 Network of detectors
4.11 Non-stationary, non-Gaussian, and non-linear data