## 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 , in particular its probability distribution. When the signal is absent the data have probability density function (pdf) , and when the signal is present the pdf is .

A full exposition of the statistical theory of signal detection that is outlined here can be found in the monographs [102569896664477]. 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 is that the signal is absent from the data and the alternative hypothesis is that the signal is present. A hypothesis test (or decision rule) is a partition of the observation set into two sets, and its complement . If data are in 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 when is true and a type II error is choosing when 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. 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 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].