The Whittle Likelihood is a frequency-based approximation to the Gaussian Likelihood which is up to a constant asymptotically efficient. The Whittle estimate is asymptotically efficient and can be interpreted as minimum distance estimate of the distance between the parametric spectral density and the (nonparametric) periodogram. It also minimises the asymptotic Kullback-Leibler divergence and, for autoregressive processes, is identical to the Yule-Walker estimate. The evaluation of the Whittle Likelihood can be done very fast by computing the periodogram via the FFT in only operations.
Suppose that a stationary, zero mean, gaussian process is observed at times . Assume has spectral density , , depending on a vector of unknown parameters . A natural approach to estimate the parameter from the sample is to maximize the likelihood function or alternatively to minimise times the log-likelihood. The later takes the form
Observe that the variance-covariance matrix can be expressed in terms of the spectral density as follows
Hence can be rewritten as
where is the Toeplitz matrix of f.
Unfortunately it is difficult to calculate the preceding function and that is especially true for the inverse of the Toeplitz matrix. However we can approximate it by using a famous result by Grenander and Szegö (1958).
Theorem (Szegö). Consider with where are the Fourier coefficients of . Then s.th. while the eigenvalues of belong to , and continuous
Note that for large , is almost a homomorphism which suggests that can be replaced by . Hence the third term of becomes
is the periodogram of the process.
This finally leads to the likelihood approximation suggested by Whittle (1953).