The test statistic used for detection is a special case of a function
of random variables. Testing the hypothesis 
using the statistics
S is a standard statistical procedure. Important examples of the
test statistics are signal variances
For white noise (
is true), the distribution of 
is a
distribution. It is remarkable that, then, 
and
are statistically independent. Note further the inequality
which is due to the extra variance,
, in the model signal with respect to the variance, 
, of
the observations. The equality in these relations holds for pure
noise.
The larger the variance of the model series 
compared to the
residual variance 
is, the more significant is the detection or
the better is the current parameter estimate, for problems (1) and (2)
respectively (Sect. 
). Usually, the test statistics S in
TSA measure a ratio of two variances. They differ according to the
models assumed and the combination of the variances chosen. Since
models depend on frequency 
(or time lag l), so do the
variances 
and 
and test statistics S.
The statistics we recommend for use in the frequency domain are the
ones introduced by Scargle and the Analysis of Variance (AOV)
statistics. These methods are implemented in MIDAS commands
SCARGLE/TSA and AOV/TSA (Sect. ), respectively.
The Scargle statistic uses a pure sine model, the AOV statistic uses a
step function (phase binning). In the time domain, we recommend to
use the 
statistic with the COVAR/TSA and
DELAY/TSA commands (Sect. ). Both COVAR/TSA and
DELAY/TSA are based on a second series of observations which is
used for the model. COVAR/TSA and DELAY/TSA differ in the
method used for the interpolation of the series: the former deploys a
step function (binning) while the latter relies on an analytical
approximation of the autocorrelation function (ACF,
Sect. ) as a more elaborate approach. Among many other
statistics we mention the one by Lafler & Kinman (1965), phase
dispersion minimization (PDM) also known as the Whittaker & Robinson
statistic (Stellingwerf, 1978), string length (Dvoretsky, 1983), and
statistic introduced by Renson (1983).
In the limit of 
(
) the sums of
squares and degrees of freedom converge and so does the variance 
(
). Since 
, increasing the number of parameters of a model 
to 
implies a decrease of 
and a corresponding decrease in the
significance of the detection. Therefore, we do not recommend to use
models (e.g. long Fourier series, fine phase binning, string length
and Renson statistics) with more parameters than are really required
for the detection of the feature in question.
In the above limits, 
and 
(
) become perfectly
correlated. Since all statistics named above except AOV use 
at least implicitly, their probability distribution may, because of
this correlation, differ considerably from what is generally supposed in the
literature (Schwarzenberg-Czerny, 1989). However, the correlation
vanishes in the asymptotic limit 
for 
,
Scargle and Whitteker & Robinson statistics, so that they yield
correct results for sufficiently large data sets. Please note that the
problem of correlation aggravates for observations with high
signal-to-noise ratio, 
, as 
, so that the statistics mentioned as using these variances
become rather insensitive.