Spectral density estimation

Remarks

Nonparametric estimators smooth the periodogram $I_n$. Let $\hat\gamma$ be the sample autocovariance, defined with the autocorrelation function. Let $w$ be a weight function with $w(0) = 1$, and let $r = r_n$ be a truncation point. A lag-window estimator has the form $$\hat f_n(\omega) = \frac{1}{2\pi} \sum_{\abs{h} \le r} w(h / r) \, \hat\gamma(h) e^{-i h \omega}.$$ Under standard conditions, such as $r_n \to \infty$ and $r_n / n \to 0$, it is consistent. Averaging periodograms of several segments of the data, as in the methods of Bartlett and Welch, is another common choice.

Parametric estimators fit an ARMA model and use its spectral density $\frac{\sigma^2}{2\pi} \abs{\theta(e^{-i\omega})}^2 / \abs{\phi(e^{-i\omega})}^2$.

Example: for a white noise with variance $\sigma^2$, the spectral density is the constant $\sigma^2 / (2\pi)$, and the averaged estimate $\frac{1}{2\pi} \cdot \frac{1}{m} \sum_j I_n(\omega_j)$ over $m$ Fourier frequencies near $\omega$ estimates it.

The rescaled periodogram $I_n(\omega) / (2\pi)$ is the simplest estimator, but it is not consistent. Under standard conditions, for $0 < \abs{\omega} < \pi$, it is asymptotically unbiased while its variance tends to $f(\omega)^2$ rather than to $0$.