Whittle likelihood

Remarks

The idea is that, for a Gaussian process with smooth spectral density, the values $I(\omega_j)$ are approximately independent and exponentially distributed with means $f_\theta(\omega_j)$. The log likelihood of independent exponential variables with these means is exactly $\ell_W(\theta)$. A version with an integral, $-\frac{n}{4\pi} \int_{-\pi}^{\pi} \left( \log f_\theta(\omega) + I(\omega) / f_\theta(\omega) \right) d\omega$, is also common. Under standard conditions the Whittle estimator is consistent and asymptotically normal with the same limit as the Gaussian maximum likelihood estimator.

Example: for white noise, $f_\theta(\omega) = \theta / (2\pi)$ with $\theta = \sigma^2 > 0$. Setting the derivative of $\ell_W(\theta) = -\sum_j \left( \log \frac{\theta}{2\pi} + \frac{2\pi I(\omega_j)}{\theta} \right)$ to zero gives $m/\theta = 2\pi \sum_j I(\omega_j) / \theta^2$, so $\hat\theta = \frac{2\pi}{m} \sum_{j=1}^{m} I(\omega_j)$.

Non-example: let $\Gamma_n = [\gamma_\theta(s - t)]_{s,t=1}^n$ be the covariance matrix of the data. The exact Gaussian log-likelihood $-\frac12 \log \det \Gamma_n - \frac12 x^T \Gamma_n^{-1} x$, up to a constant, is not the Whittle likelihood. The Whittle likelihood approximates it by treating $\Gamma_n$ as if it were diagonal in the Fourier basis.