On Kay's frequency estimator (Q2740046)

The author considers the problem of estimation of \(\omega_0\) in the model \(x_t=A\exp\{i(\omega_0t+\vartheta)\}+z_t\), where \(z_t\) is a complex Gaussian white noise. Two estimators are discussed: NEWLINE\[NEWLINE\hat\omega_0=6(N(N^2-1))^{-1}\sum_{t=0}^{N-2}(t+1)(N-1-t)\angle(x_t^*x_{t+1}),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\tilde\omega_0=6(N(N^2-1))^{-1}\angle\left(\sum_{t=0}^{N-2}(t+1)(N-1-t)(x_t^*x_{t+1})\right),NEWLINE\]NEWLINE where \(\angle a\) means the phase of \(a\). It is shown that \(\hat\omega_0\) is inconsistent whence \(\tilde\omega_0\) is consistent and \(N^{3/2}(\tilde\omega_0-\omega)\) is asymptotically normal.