The maximum likelihood method does not require that the noise in the detector is Gaussian or stationary. However, in order to derive the optimum statistic and calculate the Fisher matrix we need to know the statistical properties of the data. The probability distribution of the data may be complicated, and the derivation of the optimum statistic, the calculation of the Fisher matrix components and the false alarm probabilities may be impractical. There is however one important result that we have already mentioned. The matched-filter which is optimal for the Gaussian case is also a linear filter that gives maximum signal-to-noise ratio no matter what is the distribution of the data. Monte Carlo simulations performed by Finn [33] for the case of a network of detectors indicate that the performance of matched-filtering (i.e., the maximum likelihood method for Gaussian noise) is satisfactory for the case of non-Gaussian and stationary noise.
In the remaining part of this section we review
some statistical tests and methods to detect non-Gaussianity,
non-stationarity, and non-linearity in the data. A classical test
for a sequence of data to be Gaussian is the Kolmogorov-Smirnov
test
[23]
. It calculates the maximum distance between the cumulative
distribution of the data and that of a normal distribution, and
assesses the significance of the distance. A similar test is the
Lillifors test
[23], but it adjusts for the fact that the parameters of the normal
distribution are estimated from the data rather than specified in
advance. Another test is the Jarque-Bera test
[42]
which determines whether sample skewness and kurtosis are
unusually different from their Gaussian values.
Let
and
be two discrete in time random processes (
) and let
be independent and identically distributed (i.i.d.). We call the
process
linear
if it can be represented by
If Hypothesis 1 holds, we can test for linearity, that is, we have a second hypothesis testing problem:
If Hypothesis 4 holds, the process is linear.
Using the above tests we can
detect
non-Gaussianity and, if the process is non-Gaussian,
non-linearity of the process. The distribution of the test
statistic
, Equation (90
), can be calculated in terms of
distributions. For more details see
[38]
.
It is not difficult to examine non-stationarity of the data. One can divide the data into short segments and for each segment calculate the mean, standard deviation and estimate the spectrum. One can then investigate the variation of these quantities from one segment of the data to the other. This simple analysis can be useful in identifying and eliminating bad data. Another quantity to examine is the autocorrelation function of the data. For a stationary process the autocorrelation function should decay to zero. A test to detect certain non-stationarities used for analysis of econometric time series is the Dickey-Fuller test [21] . It models the data by an autoregressive process and it tests whether values of the parameters of the process deviate from those allowed by a stationary model. A robust test for detection non-stationarity in data from gravitational-wave detectors has been developed by Mohanty [60] . The test involves applying Student’s t-test to Fourier coefficients of segments of the data.
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