Efficient semiparametric estimation of the periods in a superposition of periodic functions with unknown shape. (English) Zbl 1150.62047
Let us assume that we have \(n\) observations of a sampled signal corrupted by Gaussian white noise, \(X(j)=s_0(jt_{n})+\varepsilon(j)\); \(1\leq j\leq n\), where \(s_0\) is an unknown real function, \(\varepsilon(j)\) are independent centred Gaussian random variables of unknown variance, and \(t_{n}\) is the sampling period. The authors consider the situation where the signal \(s_0\) can be decomposed into a sum of unknown periodic signals and are interested in the estimation of the periods of the individual periodic components. In the case of a single periodic function a consistent and asymptotically efficient semiparameteric estimator of the period is proposed. Then the authors study the case of a sum of two periodic functions of unknown shape with different periods and propose semiparamepric estimators of their periods that are consistent and asymptotically Gaussian.
Reviewer: Aleksandr D. Borisenko (Kyïv)
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
Keywords:
periodic signalsSoftware:
partsmReferences:
| [1] | DOI: 10.1111/1467-9892.00246 · Zbl 0984.62062 · doi:10.1111/1467-9892.00246 |
| [2] | Bentarzi M., Journal of Time Series Analysis 15 pp 263– (1994) |
| [3] | Bentarzi M., Econometric Theory 12 pp 88– (1996) |
| [4] | Bentarzi M., Journal of Applied Probability 35 pp 48– (1998) · Zbl 0923.62089 · doi:10.1017/S0021900200014662 |
| [5] | Franses P., Periodicity and Stochastic Trends in Economic Time Series (1996) · Zbl 0868.62088 |
| [6] | Golubev G. K., Problemy Peredachi Informatsii 24 pp 38– (1988) |
| [7] | DOI: 10.1093/biomet/87.3.545 · doi:10.1093/biomet/87.3.545 |
| [8] | Hallin M., Statistical Modeling and Analysis for Complex Data Problems pp 281– (2005) |
| [9] | DOI: 10.2307/3212772 · Zbl 0271.62122 · doi:10.2307/3212772 |
| [10] | Kavalieris L., Journal of Time Series Analysis 15 pp 613– (1994) |
| [11] | Kay S. M., IEEE Transactions on ASSP 37 pp 1987– (1989) |
| [12] | Kootsookos P. J., A review of the frequency estimation and tracking problems. Frequency estimation and tracking project (1999) |
| [13] | Kundu D., Statistical Computing: Existing Methods and Recent Developments pp 371– (2004) |
| [14] | DOI: 10.1109/TSP.2005.849156 · Zbl 1370.94164 · doi:10.1109/TSP.2005.849156 |
| [15] | Le Cam L., Asymptotic Methods in Statistical Decision Theory (1986) · Zbl 0605.62002 |
| [16] | Levy-Leduc C., Estimation semi-parametrique de la frequence de fonctions periodiques dans divers modeles statistiques: theorie et pratique (2004) |
| [17] | DOI: 10.1016/S0378-3758(00)00193-2 · Zbl 0970.62031 · doi:10.1016/S0378-3758(00)00193-2 |
| [18] | Pisarenko V. F., Geophysical Journal of the Royal Astronomical Society 10 pp 347– (1973) · doi:10.1111/j.1365-246X.1973.tb03424.x |
| [19] | Prenat M., Proceedings of the Physics in Signal and Image Processing (PSIP2001) Conference. Marseille, France (2001) |
| [20] | DOI: 10.2307/2337018 · doi:10.2307/2337018 |
| [21] | Quinn B. G., The Estimation and Tracking of Frequency. Cambridge Series in Statistical and Probabilistic Mathematics (2001) · Zbl 0969.62060 · doi:10.1017/CBO9780511609602 |
| [22] | DOI: 10.2307/2336896 · doi:10.2307/2336896 |
| [23] | DOI: 10.1109/TAP.1986.1143830 · doi:10.1109/TAP.1986.1143830 |
| [24] | van der Vaart A. W., Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics (1998) · Zbl 0910.62001 |
| [25] | DOI: 10.2307/2334314 · doi:10.2307/2334314 |
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