A statistically and computationally efficient method for frequency estimation (Q1411875)

Let \(\{\varepsilon _t\}\) be a zero-mean stationary ergodic process. Define \(y_t=\beta \cos (\omega _0 t+\phi)+ \varepsilon _t\), \(t=1,\dots ,n\), where \(\beta >0\), \(\omega _0\in (0,\pi)\), and \(\phi \) is a uniformly distributed random variable in \((-\pi , \pi ]\). For any given \(\eta \in (0,1)\) and \[ \alpha =\cos \omega \in (-2\eta (1+\eta ^2)^{-1}, 2\eta (1+\eta ^2)^{-1}), \] define \(\{y_t(\alpha)\}\) recursively by \[ y_t(\alpha)+2\theta (\alpha)\eta y_{t-1}(\alpha) +\eta ^2 y_{t-2}(\alpha)=y_t, \] with \(y_{-1}(\alpha)=y_0(\alpha)=0\) and \(\theta (\alpha)= -\alpha (1+\eta ^2)/(2\eta)\). Then \[ \rho _n(\alpha) =(1+\eta ^2)^{-1} \left [\sum _{t=1}^n y_{t-1}^2 (\alpha)\right ] ^{-1} \sum _{t=1}^n y_{t-1}(\alpha)\{y_t(\alpha)+ \eta ^2 y_{t-2}(\alpha)\} \] is an estimate of the lag-one autocorrelation coefficient of \(\{y_t(\alpha)\}\). The contraction mapping (CM) method is based on the procedure \(\widehat {\alpha }_n^{(m)} =\rho _n(\widehat {\alpha }_n^{(m-1)})\) and a frequency estimator is \(\widehat {\omega }_n=\text{arc cos} \widehat {\alpha }_n\) where \(\widehat {\alpha }_n= \lim \widehat {\alpha }_n^{(m)}\). The authors establish connections between the initial guess and the precision of the CM estimator \(\widehat {\omega }_n\) of \(\omega _0\) and construct an algorithm with varying \(\eta \) adaptable to possibly poor initial values. This algorithm can accommodate initial values of precision \({\mathcal O}(1)\) and converges to a final estimator whose precision is arbitrary close to the optimal rate \({\mathcal O}(n^{-3/2})\). It is also proved that the estimator is asymptotically normally distributed.