Corrections to the probability distribution

In principle, it is possible to compute a value of the statistic
for a single frequency
and to test its consistency
with a random signal (*H*_{o}). The common procedure of inspecting the
whole periodogram for a detected signal corresponds to the *N*-fold
repetition of the single test for a set of trial frequencies,
.
The probability of the whole periodogram being
consistent with *H*_{o} is
for
.
The factor *N* means that there is an increased probability of
accepting a given value of the statistic as consistent with a random
signal. Therefore, increasing the number of trial frequencies
decreases the sensitivity for the detection of a significant signal
and accordingly is called the penalty factor for multiple trials or
for the frequency bandwidth used. The true number of independent
frequencies, *N*_{t}, remains generally unknown. It is usually less than the
number of resolved frequencies
(Sect.
12.3.1) because of aliasing and still less than the number of
computed frequencies *N*_{c}, because of oversampling:
.
For a practical and conservative estimate, we recommend to use
*N*_{r} as the number of trial frequencies, *N*.

According to the standard null hypothesis, *H*_{o}, the noise is white
noise. This is not the case in many practical cases. For instance,
often the noise is a stochastic process with a certain correlation
length
*l*_{corr}>0, so that on average *n*_{corr} consecutive
observations are correlated. Such noise corresponds to white noise
passed through a low pass filter which cuts off all frequencies above
1/*l*_{corr}. Such correlation is not usually taken into account by
standard test statistics. The effect of this correlation is to reduce
the effective number of observations by a factor *n*_{corr}(Schwarzenberg-Czerny, 1989). This has to be accounted for by scaling
both the statistics *S* and the number of its degrees of freedom *n*_{j}by factors depending on *n*_{corr}.

In the test statistic, a continuum level which is inconsistent with
the expected value of the statistic
may indicate the presence
of such a correlation between consecutive data points. A practical
recipe to measure the correlation is to compute the residual time
series (e.g. with the `SINEFIT/TSA` command) and to look for its
correlation length with `COVAR/TSA` command. The effect of the
correlation in the parameter estimation is an underestimation of the
uncertainties of the parameters; the true variances of the parameters
are a factor *n*_{corr} larger than computed.

In the command individual descriptions, we often refer to probability
distributions of specific statistics. For the properties of these
individual distributions see e.g. Eadie *et. al.* (1971), Brandt
(1970), and Abramovitz & Stegun (1972). The two latter references
contain tables. For a computer code for the computation of the
cumulative probabilities see Press *et. al.* (1986).