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<H2><A NAME="SECTION001723000000000000000">
Test statistics</A>
</H2>
<P>
The test statistic used for detection is a special case of a function
of random variables. Testing the hypothesis <I>H</I><SUB><I>o</I></SUB> using the statistics
<I>S</I> is a standard statistical procedure. Important examples of the
test statistics are signal variances
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
Var_j \equiv Var[X^{(j)}] =
{1\over{n_j}} \sum_{k=1}^{n_o}\left(x^{(j)}_k\right)^2 ~~~j=o,m,r
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="455" HEIGHT="57"
SRC="img447.gif"
ALT="\begin{displaymath}Var_j \equiv Var[X^{(j)}] =
{1\over{n_j}} \sum_{k=1}^{n_o}\left(x^{(j)}_k\right)^2 ~~~j=o,m,r
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.1)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
For white noise (<I>H</I><SUB><I>o</I></SUB> is true), the distribution of <I>Var</I><SUB><I>j</I></SUB> is a
<!-- MATH: $\chi^2(n_j)$ -->
<IMG
WIDTH="22" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img448.gif"
ALT="$\chi^2(n_j)$">
distribution. It is remarkable that, then, <I>Var</I><SUB><I>m</I></SUB> and
<I>Var</I><SUB><I>r</I></SUB> are statistically independent. Note further the inequality
<!-- MATH: $Var_m\geq Var_o\geq Var_r$ -->
<IMG
WIDTH="170" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img449.gif"
ALT="$Var_m\geq Var_o\geq Var_r$">
which is due to the extra variance,
<I>Var</I><SUB><I>m</I></SUB>, in the model signal with respect to the variance, <I>Var</I><SUB><I>o</I></SUB>, of
the observations. The equality in these relations holds for pure
noise.
<P>
The larger the variance of the model series <I>Var</I><SUB><I>m</I></SUB> compared to the
residual variance <I>Var</I><SUB><I>r</I></SUB> is, the more significant is the detection or
the better is the current parameter estimate, for problems (1) and (2)
respectively (Sect. <A HREF="node222.html#s:gen">12.2</A>). Usually, the test statistics <I>S</I> 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 <IMG
WIDTH="42" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img450.gif"
ALT="$\nu$">
(or time lag <I>l</I>), so do the
variances <I>Var</I><SUB><I>m</I></SUB> and <I>Var</I><SUB><I>r</I></SUB> and test statistics <I>S</I>.
<P>
The statistics we recommend for use in the frequency domain are the
ones introduced by Scargle and the Analysis of Variance (AOV)
statistics. These statistics are used in the MIDAS commands <TT>SCARGLE/TSA, ORT/TSA</TT> and <TT>AOV/TSA</TT> (Sect. <A HREF="node241.html#s:freq">12.4.6</A>).
The SCARGLE/TSA command uses a pure sine model, the ORT/TSA uses Fourier
series and the AOV/TSA uses a step function (phase binning).
In the time domain, we recommend to use the
<!-- MATH: $Var_r\equiv \chi^2$ -->
<IMG
WIDTH="61" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
SRC="img451.gif"
ALT="$Var_r\equiv \chi^2$">
statistic with the <TT>COVAR/TSA</TT> and
<TT>DELAY/TSA</TT> commands (Sect. <A HREF="node242.html#s:tim">12.4.7</A>). Both <TT>COVAR/TSA</TT> and
<TT>DELAY/TSA</TT> are based on a second series of observations which is
used for the model. <TT>COVAR/TSA</TT> and <TT>DELAY/TSA</TT> 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. <A HREF="node233.html#s:psac">12.3.2</A>) 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).
<P>
In the limit of
<!-- MATH: $n_r\rightarrow 0$ -->
<IMG
WIDTH="28" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img452.gif"
ALT="$n_r\rightarrow 0$">
(
<!-- MATH: $n_m\rightarrow n_o$ -->
<IMG
WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img453.gif"
ALT="$n_m\rightarrow n_o$">)
the sums of
squares and degrees of freedom converge and so does the variance
<!-- MATH: $Var_r
\rightarrow Var_o$ -->
<IMG
WIDTH="90" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img454.gif"
ALT="$Var_r
\rightarrow Var_o$">
(
<!-- MATH: $Var_m \rightarrow Var_o$ -->
<IMG
WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img455.gif"
ALT="$Var_m \rightarrow Var_o$">). Since
<!-- MATH: $Var_m\geq
Var_o$ -->
<IMG
WIDTH="92" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img456.gif"
ALT="$Var_m\geq
Var_o$">,
increasing the number of parameters of a model <I>n</I><SUB><I>m</I></SUB> to <I>n</I><SUB><I>o</I></SUB>implies a decrease of <I>Var</I><SUB><I>m</I></SUB> 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.
<P>
In the above limits, <I>Var</I><SUB><I>o</I></SUB> and <I>Var</I><SUB><I>r</I></SUB> (<I>Var</I><SUB><I>m</I></SUB>) become perfectly
correlated. Since all statistics named above except AOV use <I>Var</I><SUB><I>o</I></SUB>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
<!-- MATH: $n_o\rightarrow\infty$ -->
<IMG
WIDTH="39" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
SRC="img457.gif"
ALT="$n_o\rightarrow\infty$">
for <IMG
WIDTH="54" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
SRC="img458.gif"
ALT="$\chi^2$">,
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,
<!-- MATH: $S/N \rightarrow \infty$ -->
<IMG
WIDTH="60" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img459.gif"
ALT="$S/N \rightarrow \infty$">,
as
<!-- MATH: $Var_m \rightarrow
Var_o$ -->
<IMG
WIDTH="98" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img460.gif"
ALT="$Var_m \rightarrow
Var_o$">,
so that the statistics mentioned as using these variances
become rather insensitive.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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