Estimating the spectrum
We now consider the quality of 
as an estimate of 
. We
have already shown that
However we will see that the variance of this estimate 
of f does
not go to 0 so that the estimate is not consistent.
It is easier technically to consider the case of a normal mean 0
process X. For
normal data the real and imaginary parts of 
have normal distributions.
Both have mean 0. The variances are
and
while the covariance between the real and imginary parts is
Consider as an example the covariance, and use the usual complex exponential identities to write the covariance as
Now make the change of variables u=r-s and v=r+s in the double sum.
The variable u runs from 
to T-1 while when u is fixed the
possible values of v run, for u positive from u to 
by
increments of 2 and, for u negative from -u to 
by increments
of 2. For each value of u there are then T-|u| possible values of v
and the covariance becomes
The last two terms, involving u only, are
The terms u and -u cancel each other while the term with u=0 is 0 itself so that this term is 0.
The terms above involving v may be simplified by using geometric series
to do the inside sums over v. The result is a coefficient of 
which is bounded (bounded by 
for instance. Then
since
we have checked that the covariance between the real and imaginary parts of
converges to 0 as 
.
Our previous calculations of the expectation of 
can be mimicked to show that the two variances each converge to
. It follows that the vector 
converges to a bivariate standard normal.
The squared length of this vector then converges in distribution to
the squared length of a standard bivariate normal which is exactly
or exponentail with mean 2.
Summary: 
converges in distribution to an
exponential random variable with mean 
. In particular,
is not a consistent estimator of 
.
Improved estimates
To get better estimates we need either to resort to parametric estimation
techniques or do some smoothing. We will look at the latter idea first.
If 
is smooth in the neighbourhood of some 
then we
can take estimates of 
at a number of points nearby to
and average them somehow. Averaging will reduce the variance
though it will introduce bias usually because the things being averaged all
have different expected values.
The simplest kind of estimator is a moving average -- we define
It turns out that the quantities being averaged are asymptotically
independent so that the estimate has the same distribution as an average
of 2L+1 exponentials which is just a chi-squared with L+2 degrees
of freedom multiplied by 
. It is possible then to
produce a consistent estimate by letting L grow slowly with T but we
won't investigate this rather mathematical problem carefully here.
Other weighted averages are possible; several are implemented in the SPlus function spectrum. Here are some points to note about this estimation problem:
has expected value
where a (complicated) formula for the
bias can be deduced from the algebra above. The expected value of
an estimate of the form
is then
If f is roughly linear around 
then the first term will
be quite close to 
when the weights make the estimate an
average, that is, they sum to 1. However, this approximation will be poor
in the neighbourhood of any peak in the spectrum which will be flattened
by this averaging. The second term in the expectation, on the other hand,
has no particular reason to average out to 0; increasing L without
dealing with this bias will eventually be fruitless as the bias becomes
the dominant component in the error. A common tactic to dealing with this
bias is tapering, where we compute
and use as a periodogram
where the tapering function h typically decreases to 0 at 0 and at 1.
is a
filter such that 
is nearly white noise then we could
. Call the resulting estimate
.
by
where A is the frequency response function of the filter 
.
Here are several spectral estimates for the spectrum of the sunspots series:
