The question which method of the detection of features in the test
statistic is the most sensitive, is of considerable importance for all
practical TSA applications. It directly translates into the comparison
of the power of different statistics. Let *Y* be a signal of some
physical meaning, i.e. different from the pure noise. A power of the test statistic is the probability that *H*_{o} is accepted for
the given *Y*:
.
In other words, this is the
probability of the false rejection of the no-noise signal *Y* as pure
noise. Generally, the relative power of various statistics depends on
the type of the signal *Y*. An important practical consequence of this
fact is that there is no such thing as the universally best method
for time series analysis. A method that is good for one type of
signals may be poor for another one. An extensive list of signal types
and recommended methods is clearly out of scope of the present manual.
However, in order to guide the user's intuition we quote below the
results of a comparison of the powers of various test statistics for
two types of oscillations, both of small amplitude and observed at
more or less even intervals (Schwarzenberg-Czerny, 1989):

Let us first consider a sinusoidal oscillation with an amplitude not
exceeding the noise. Then, all statistics based on models with 3
parameters have similar values of the test power: power spectrum,
Scargle statistics,
fit of a sinusoid, ORT(1), AOV(3)
and corrected PDM(3) statistics. (Please
note that we depart here from the conventional notation by indicating
in brackets the number *n*_{m} of the degrees of freedom of the model -
e.g. the number of series terms or phase bins - instead of the number
of the residual degrees of freedom *n*_{r}.) Statistics with more than
3 parameters, e.g. ORT(*n*/2+1), AOV(*n*), PDM(*n*) and
with *n*>3 and
an extended Fourier series have less power. Our final choice is guided
by the availability of the analytical probability distribution of the
test statistics. Summing up, we recommend tu use for the detection of sinusoidal
and other smooth oscillations of small amplitude the statistics with
a coarse phase resolution, e.g. ORT(1), Scargle and AOV(3), .

For a narrow gaussian pulse or eclipse of width *w* repeating with
period *P* the most powerful statistics are these with the matching
resolution: ORT(*P*/2*w*), AOV(*P*/*w*) and
P/w). Power spectrum, Scargle, ORT(1), AOV(3), ,
ORT(n/2), AOV(n) and ,
all have less power. Note the equivalence
of the
and Scargle's statistic (Lomb, 1976, Scargle, 1982)
and the near-equivalence of the power spectrum and Scargle's
statistics in the case of nearly uniformly sampled observations.
Considering both test power and computational convenience we recommend
for signals with sharp features, e.g. narrow pulses or eclipses, to
use the ORT and AOV with the resolution matched to the
width of these features.