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  "body": "\n## Definition\n\nLet $x_1, \\dots, x_n$ be observations of a weakly [[stationary process|DT0QXJE705]] with mean $0$.\nLet $\\set{f_\\theta : \\theta \\in \\Theta}$ be a parametric family of spectral densities on $[-\\pi, \\pi]$, with $f_\\theta \u003e 0$.\nLet $I(\\omega) = \\frac{1}{2\\pi n} \\abs{\\sum_{t=1}^n x_t e^{-i t \\omega}}^2$ be the [[periodogram|3ZYM0HXA3E]], scaled to estimate the spectral density, and let $\\omega_j = 2\\pi j / n$ for $j = 1, \\dots, m$ with $m = \\floor{(n-1)/2}$.\nThe *Whittle log-likelihood* is\n$$\\ell_W(\\theta) = -\\sum_{j=1}^{m} \\left( \\log f_\\theta(\\omega_j) + \\frac{I(\\omega_j)}{f_\\theta(\\omega_j)} \\right).$$\nThe *Whittle estimator* is a maximizer $\\hat\\theta$ of $\\ell_W$ over $\\Theta$.\n\n## Remarks\n\nThe idea is that, for a Gaussian process with smooth spectral density, the values $I(\\omega_j)$ are approximately independent and exponentially distributed with means $f_\\theta(\\omega_j)$.\nThe log [[likelihood|0VW9E4QXR6]] of independent exponential variables with these means is exactly $\\ell_W(\\theta)$.\nA 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.\nUnder standard conditions the Whittle estimator is consistent and asymptotically normal with the same limit as the Gaussian [[maximum likelihood|0FREPFKNS7]] estimator.\n\nExample: for white noise, $f_\\theta(\\omega) = \\theta / (2\\pi)$ with $\\theta = \\sigma^2 \u003e 0$.\nSetting 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 $\\hat\\theta = \\frac{2\\pi}{m} \\sum_{j=1}^{m} I(\\omega_j)$.\n\nNon-example: let $\\Gamma_n = [\\gamma_\\theta(s - t)]_{s,t=1}^n$ be the covariance matrix of the data.\nThe 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.\nThe Whittle likelihood approximates it by treating $\\Gamma_n$ as if it were diagonal in the Fourier basis.\n",
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