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3 Statistical Theory of Signal Detection

The gravitational-wave signal will be buried in the noise of the detector and the data from the detector will be a random process. Consequently the problem of extracting the signal from the noise is a statistical one. The basic idea behind the signal detection is that the presence of the signal changes the statistical characteristics of the data x, in particular its probability distribution. When the signal is absent the data have probability density function (pdf) p0(x ), and when the signal is present the pdf is p1(x).

A full exposition of the statistical theory of signal detection that is outlined here can be found in the monographs [102569896Jump To The Next Citation Point6644Jump To The Next Citation Point77]. A general introduction to stochastic processes is given in [100]. Advanced treatment of the subject can be found in [64101].

The problem of detecting the signal in noise can be posed as a statistical hypothesis testing problem. The null hypothesis H0 is that the signal is absent from the data and the alternative hypothesis H1 is that the signal is present. A hypothesis test (or decision rule) δ is a partition of the observation set into two sets, ℛ and its complement ℛ ′. If data are in ℛ we accept the null hypothesis, otherwise we reject it. There are two kinds of errors that we can make. A type I error is choosing hypothesis H 1 when H0 is true and a type II error is choosing H0 when H1 is true. In signal detection theory the probability of a type I error is called the false alarm probability, whereas the probability of a type II error is called the false dismissal probability. 1 − (false dismissal probability) is the probability of detection of the signal. In hypothesis testing the probability of a type I error is called the significance of the test, whereas 1 − (probability of type II error) is called the power of the test.

The problem is to find a test that is in some way optimal. There are several approaches to find such a test. The subject is covered in detail in many books on statistics, for example see references [544162].

 3.1 Bayesian approach
 3.2 Minimax approach
 3.3 Neyman–Pearson approach
 3.4 Likelihood ratio test
  3.4.1 Gaussian case – The matched filter

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