The Annals of Mathematical Statistics

Estimation of the Spectrum

V. K. Murthy

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This paper extends some results of Grenander [1] relating to discrete real stationary normal processes with absolutely continuous spectrum to the case in which the spectrum also contains a step function with a finite number of salt uses. It is shown by Grenander [1] that the periodogram is an asymptotically unbiased estimate of the spectral density $f(\lambda)$ and that its variance is $\lbrack f(\lambda)\rbrack^2$ or $2\lbrack f(\lambda)\rbrack^2$, according as $\lambda \neq 0$ or $\lambda = 0$. In the present paper the same results are established at a point of continuity. The consistency of a suitably weighted periodogram for estimating $f(\lambda)$ is established by Grenander [1]. In this paper a weighted periodogram estimate similar to that of Grenander (except that the weight function is more restricted) is constructed which consistently estimates the spectral density at a point of continuity. It appears that this extended result leads to a direct approach to the location of a single periodicity irrespective of the presence of others in the time series.

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Ann. Math. Statist., Volume 32, Number 3 (1961), 730-738.

First available in Project Euclid: 27 April 2007

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Murthy, V. K. Estimation of the Spectrum. Ann. Math. Statist. 32 (1961), no. 3, 730--738. doi:10.1214/aoms/1177704968.

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