On autocorrelation estimation in mixed-spectrum Gaussian processes
Let \(z = \{z_ t\}\), \(t = 0,\pm 1,\dots\) be a real-valued, zero-mean stationary Gaussian process with autocorrelation \(\rho_ z(k)\), and let \(d = \{d_ t\}\) be the associated zero-crossing indicator process. Given \(z_ 1,\dots,z_ N\), the empirical zero-crossing rate is defined as \(\widehat{\gamma} = (\pi/(N - 1))\sum^ N_{t = 2} d_ t\). Then \(\rho_ z(1) = \cos(E(\widehat{\gamma}))\). The paper considers the estimation of \(\rho_ z(1)\), using \(\widehat{\rho}_ z(1) = \cos(\widehat{\gamma})\). It is proved that it is consistent for a spectrum supported at a single frequency, or a spectrum for which the signal and noise have the exact same first-order autocorrelation.
real- valued, zero-mean stationary Gaussian process
scientific article; zbMATH DE number 519769
scientific article