Exact PDF of an ARMA process  Efficient method using Levinson algorithm
The exact PDF of an ARMA process can be written as in (9.6)
where is the symmetric Toeplitz
covariance matrix of an ARMA process
formed from (9.15).
Computing , and even just storing it, for large is inefficient.
Using the Levinson algorithm, an efficient implementation of the exact PDF may be obtained.
When is large, the computing the term
is prohibitive
and when is very large, even storing is inefficient.
The problem is solved efficiently using the Levinson algorithm.
Let x be an Nby1 vector and let be the symmetric NbyN Toeplitz matrix
formed from the Nby1 ACF vector r. The function
[y,ldetR] = toepsolve(r,x,N);
is the same as
R = toeplitz(r(1:N));
y = R \ x;
ldetR = log(det(R));
We then have
The execution time of
software/toepsolve.m is much faster (see Figure 9.3)
and includes as a byproduct the determinant of R.
As predicted by theory, the standard approach is order , while Levinson is .
Figure 9.3:
Comparison of execution times of y = R b versus y = toepsolve(r,b,N) as
a function of N.

The exact PDF can be therefore computed by
[y,ldetR] = toepsolve(r,x,N);
lpx = N/2*log(2*pi) .5*ldetR .5*x'*y;
Baggenstoss
20170519