##
**On frequency estimation.**
*(English)*
Zbl 0654.62077

A time series model of the form
\[
X_ t=a\cos (wt+f)+e_ t
\]
where \(e_ t\) is a stationary noise sequence is considered. The paper discusses a least squares procedure for estimating a, w and f of the harmonic component.

One aim of the paper is to see how reliable the asymptotic theory of the estimates of w is. It was found that the product of the amplitude and the sample size, n, must be quite large in order for the asymptotic theory to be meaningful. Otherwise the frequency estimate is much more variable than indicated by the asymptotic theory and the amplitude estimate is biased. This suggests that a general application of the asymptotic results can be quite misleading when there are small peaks in the periodogram (small amplitude).

The paper is also concerned with computational issues of solving the nonlinear estimation equations. The numerical problems increase with sample size n, there are many local minima and the convergence of the iterative search is extremely sensitive to the starting values.

One aim of the paper is to see how reliable the asymptotic theory of the estimates of w is. It was found that the product of the amplitude and the sample size, n, must be quite large in order for the asymptotic theory to be meaningful. Otherwise the frequency estimate is much more variable than indicated by the asymptotic theory and the amplitude estimate is biased. This suggests that a general application of the asymptotic results can be quite misleading when there are small peaks in the periodogram (small amplitude).

The paper is also concerned with computational issues of solving the nonlinear estimation equations. The numerical problems increase with sample size n, there are many local minima and the convergence of the iterative search is extremely sensitive to the starting values.

Reviewer: D.Rasch

### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62F12 | Asymptotic properties of parametric estimators |

62M15 | Inference from stochastic processes and spectral analysis |