The Annals of Mathematical Statistics
- Ann. Math. Statist.
- Volume 32, Number 3 (1961), 730-738.
Estimation of the Spectrum
This paper extends some results of Grenander  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  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 . 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.
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. https://projecteuclid.org/euclid.aoms/1177704968