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  "body": "\n## Definition\n\nLet $(X_t)_{t \\in \\Z}$ be a real weakly [[stationary process|DT0QXJE705]] with autocovariance $\\gamma$ satisfying $\\sum_{h} \\abs{\\gamma(h)} \u003c \\infty$.\nIts *spectral density* is\n$$f(\\omega) = \\frac{1}{2\\pi} \\sum_{h \\in \\Z} \\gamma(h) e^{-i h \\omega}, \\qquad \\omega \\in [-\\pi, \\pi],$$\nso that $\\gamma(h) = \\int_{-\\pi}^{\\pi} e^{i h \\omega} f(\\omega) \\, d\\omega$.\n*Spectral density estimation* is the construction of estimators $\\hat f_n(\\omega)$ of $f(\\omega)$ from observations $X_1, \\dots, X_n$.\nSuch an estimator is a function of the data that should be close to $f$ for large $n$.\n\n## Remarks\n\nNonparametric estimators smooth the [[periodogram|3ZYM0HXA3E]] $I_n$.\nLet $\\hat\\gamma$ be the sample autocovariance, defined with the [[autocorrelation function|YCZHYY8VE5]].\nLet $w$ be a weight function with $w(0) = 1$, and let $r = r_n$ be a truncation point.\nA lag-window estimator has the form\n$$\\hat f_n(\\omega) = \\frac{1}{2\\pi} \\sum_{\\abs{h} \\le r} w(h / r) \\, \\hat\\gamma(h) e^{-i h \\omega}.$$\nUnder standard conditions, such as $r_n \\to \\infty$ and $r_n / n \\to 0$, it is consistent.\nAveraging periodograms of several segments of the data, as in the methods of Bartlett and Welch, is another common choice.\n\nParametric estimators fit an [[ARMA model|B1HNX0YT0Z]] and use its spectral density $\\frac{\\sigma^2}{2\\pi} \\abs{\\theta(e^{-i\\omega})}^2 / \\abs{\\phi(e^{-i\\omega})}^2$.\n\nExample: 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.\n\nThe rescaled periodogram $I_n(\\omega) / (2\\pi)$ is the simplest estimator, but it is not consistent.\nUnder standard conditions, for $0 \u003c \\abs{\\omega} \u003c \\pi$, it is asymptotically unbiased while its variance tends to $f(\\omega)^2$ rather than to $0$.\n",
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