## 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