##

CR Bound analysis of the exact PDF of an ARMA processes - filtering approach.

The results of section 9.2.6 are still a little bit cumbersome to
implement.
Using filtering, we may find a much more efficient approach to finding the
derivatives of the exact ARMA PDF (9.6).
We can rewrite (9.27) as

In the time domain, this suggests a correlation between `u` and `v` where
u = filter(a,b,x)/sqrt(sig2);
v = filter(1,b,x)/sqrt(sig2);
for m=1:P,
da(m)= -sum(u(K+1:N) .* v(K+1-m:N-m));
end;

where is the impulse response length of the
wihtening filter (Section 9.2.3).
Similarly, we can rewrite (9.28) as

In the time domain, this suggests a correlation between `u` and `w`, where
u = filter(a,b,x)/sqrt(sig2);
w=filter(1,b,u);
for m=1:Q,
db(m)= sum(u(K+1:N) .* w(K+1-m:N-m));
end;

Of course, this approach, however, misses the first elements in the summation.
In fact, we can patch it up.
If we use Section 9.2.2 for the first elements only,
then add in the above values, we can obtain an exact hybrid approach based on filtering
with order computation. See `software/pdf_arma_exact.m`.

Baggenstoss
2017-05-19