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 (
). 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 
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, 
, remains generally unknown. It is usually less than the
number of resolved frequencies 
(Sect.
) because of aliasing and still less than the number of
computed frequencies 
, because of oversampling: 
. For a practical and conservative estimate, we recommend to use
as the number of trial frequencies, N.
According to the standard null hypothesis, 
, 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 
, so that on average 
consecutive
observations are correlated. Such noise corresponds to white noise
passed through a low pass filter which cuts off all frequencies above
. 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 
(Schwarzenberg-Czerny, 1989). This has to be accounted for by scaling
both the statistics S and the number of its degrees of freedom 
by factors depending on 
.
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 
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).