## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 28, Number 2 (1957), 329-348.

### On Consistent Estimates of the Spectrum of a Stationary Time Series

#### Abstract

This paper is concerned with the spectral analysis of wide sense stationary time series which possess a spectral density function and whose fourth moment functions satisfy an integrability condition (which includes Gaussian processes). Consistent estimates are obtained for the spectral density function as well as for the spectral distribution function and a general class of spectral averages. Optimum consistent estimates are chosen on the basis of criteria involving the notions of order of consistency and asymptotic variance. The problem of interpolating the estimated spectral density, so that only a finite number of quantities need be computed to determine the entire graph, is also discussed. Both continuous and discrete time series are treated.

#### Article information

**Source**

Ann. Math. Statist., Volume 28, Number 2 (1957), 329-348.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706962

**Digital Object Identifier**

doi:10.1214/aoms/1177706962

**Mathematical Reviews number (MathSciNet)**

MR88833

**Zentralblatt MATH identifier**

0081.14102

**JSTOR**

links.jstor.org

#### Citation

Parzen, Emanuel. On Consistent Estimates of the Spectrum of a Stationary Time Series. Ann. Math. Statist. 28 (1957), no. 2, 329--348. doi:10.1214/aoms/1177706962. https://projecteuclid.org/euclid.aoms/1177706962