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Consistent estimates for hidden frequencies in a linear process

Published online by Cambridge University Press:  01 July 2016

Chen Zhao-Guo
Chen Zhao-Guo*
Affiliation:
Institute of Applied Mathematics, Chinese Academy of Sciences
*
∗Postal address: Institute of Applied Mathematics, Academia Sinica, Beijing, China.
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Abstract

Traditional methods for detecting the hidden frequencies in white or coloured noise is based on the distribution of periodogram ordinates Ι (2πj/N) and hypothesis testing. The power is low when a hidden frequency falls midway between two 2nj/N and it is difficult to discuss the consistency of the procedures. Using theorems on almost sure convergence of the periodogram, this paper offers a procedure for detecting and estimating the hidden frequencies, and discusses the consistency and the rates of convergence. Simulation shows very good results, when the spectrum of the noise is not too steep.

Keywords

MAXIMUM OF THE PERIODOGRAMFEJER KERNELSTRONG CONSISTENCYLAW OF THE ITERATED LOGARITHMSECONDARY ANALYSIS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 295 - 314
Copyright
Copyright © Applied Probability Trust 1988 

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References

Hong-Zhi, An, Zhao-Guo, Chen and Hannan, E. J. (1983) The maximum of periodogram. J. Multivariate Anal. 13, 383–400.Google Scholar
Fisher, A. R. (1929) Test of significance in harmonic analysis. Proc. R. Soc. London A 125, 54–59.Google Scholar
Grenander, U. and Rosenblatt, M. (1957) Statistical Analysis of Stationary Time Series. Wiley, New York.Google Scholar
Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Hannan, E. J. (1961) Testing for a jump in the spectral function. J. R. Statist. Soc. B 23, 394–404.Google Scholar
Hannan, E. J. and Mackisack, M. (1986) A law of the iterated logarithm for an estimate of frequency. Stoch. Proc. Appl. 22, 103–109.Google Scholar
Hartley, H. O. (1949) Tests of significance in harmonic analysis. Biometrika 36, 194–201.CrossRefGoogle ScholarPubMed
Shuyuan, He (1984) On estimating the hidden periodicities in linear time series models. M.S. Thesis, Beijing University. Extended abstract in China-Japan Symposium on Statistics, Beijing, China.Google Scholar
Heyde, C. C. and Scott, D. J. (1973) Invariance principle for the law of the iterated logarithm for martingales and processes with stationary increments. Ann. Prob. 1, 428–436.Google Scholar
Mardia, K. V. (1972) Statistics of Directional Data. Academic Press, New York.Google Scholar
Priestley, M. B. (1981) Spectral Analysis and Time Series, Vol. 1. Academic Press, New York.Google Scholar
Turkman, K. F. and Walker, A. M. (1984) On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients. Adv. Appl. Prob. 16, 819–842.Google Scholar
Jiagang, Wang (1983) Some estimation problems for linear model of time series. Chinese Ann. Math. A4, 19–35.Google Scholar
Xiaobao, Wang (1985) Law of the iterated logarithm for the periodogram. Chinese J. Appl. Prob. Statist. 1, 148–154.Google Scholar
Xiaobao, Wang (1988) Law of the iterated logarithm for estimates of frequencies. Chinese J. Appl. Prob. Statist. To appear.Google Scholar
Whittle, P. (1952) The simultaneous estimation of a time series harmonic components and covariance structure. Trab. Estadist. 3, 43–57.Google Scholar