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The SPECTRA Procedure |

PROC SPECTRA prints two test statistics for white noise when the WHITETEST option is specified: Fisher's Kappa (Davis 1941, Fuller 1976) and Bartlett's Kolmogorov-Smirnov statistic (Bartlett 1966, Fuller 1976, Durbin 1967).

If the time series is a sequence of independent
random variables with mean 0 and variance ,
then the periodogram, *J*_{k}, will have
the same expected value for all *k*.
For a time series with nonzero autocorrelation,
each ordinate of the periodogram, *J*_{k}, will have different
expected values. The Fisher's Kappa statistic tests whether
the largest *J*_{k} can be considered different from
the mean of the *J*_{k}. Critical values
for the Fisher's Kappa test can be found in Fuller 1976 and
*SAS/ETS Software: Applications Guide 1*.

The Kolmogorov-Smirnov statistic reported by PROC SPECTRA
has the same asymtotic distribution as Bartlett's test (Durbin 1967).
The Kolmogorov-Smirnov statistic compares the
normalized cumulative periodogram with the cumulative distribution function
of a uniform(0,1) random variable.
The normalized cumulative periodogram, *F*_{j}, of the series is

where *m* = [*n*/2] if *n* is even or
*m* = [(*n*-1)/2] if *n* is odd.
The test statistic is the maximum absolute difference of the
normalized cumulative periodogram and the uniform cumulative
distribution function. For *m*-1 greater than 100, if
Bartlett's Kolmogorov-Smirnov statistic exceeds the critical value

where *a*=1.36 or *a*=1.63 corresponding to 5% or 1% significance
levels respectively, then reject the null hypothesis that the series
represents white noise. Critical values for *m*-1 < 100 can
be found in a table of significance points of the Kolmogorov-Smirnov
statistics with sample size *m*-1 (Miller 1956, Owen 1962).

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