Spectral methods for small sample time series: a complete periodogram approach

Das, Sourav, Subba Rao, Suhasini, and Yang, Junho (2021) Spectral methods for small sample time series: a complete periodogram approach. Journal of Time Series Analysis. (In Press)

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Abstract

The periodogram is a widely used tool to analyze second order stationary time series. An attractive feature of the periodogram is that the expectation of the periodogram is approximately equal to the underlying spectral density of the time series. However, this is only an approximation, and it is well known that the periodogram has a finite sample bias, which can be severe in small samples. In this article, we show that the bias arises because of the finite boundary of observation in one of the discrete Fourier transforms which is used in the construction of the periodogram. Moreover, we show that by using the best linear predictors of the time series over the boundary of observation we can obtain a ‘complete periodogram’ that is an unbiased estimator of the spectral density. In practice, the ‘complete periodogram’ cannot be evaluated as the best linear predictors are unknown. We propose a method for estimating the best linear predictors and prove that the resulting ‘estimated complete periodogram’ has a smaller bias than the regular periodogram. The estimated complete periodogram and a tapered version of it are used to estimate parameters, which can be represented in terms of the integrated spectral density. We prove that the resulting estimators have a smaller bias than their regular periodogram counterparts. The proposed method is illustrated with simulations and real data.

Item ID: 67528
Item Type: Article (Research - C1)
ISSN: 1467-9892
Keywords: data taper; discrete Fourier transform; periodogram; prediction and second order stationary time series
Copyright Information: © 2021 John Wiley & Sons Ltd
Funders: National Science Foundation (NSF), James Cook University (JCU)
Projects and Grants: NSF grant DMS-1812054
Date Deposited: 06 Jul 2021 05:05
FoR Codes: 41 ENVIRONMENTAL SCIENCES > 4199 Other environmental sciences > 419999 Other environmental sciences not elsewhere classified @ 10%
49 MATHEMATICAL SCIENCES > 4905 Statistics > 490511 Time series and spatial modelling @ 45%
49 MATHEMATICAL SCIENCES > 4905 Statistics > 490509 Statistical theory @ 45%
SEO Codes: 24 MANUFACTURING > 2407 Fabricated metal products > 240799 Fabricated metal products not elsewhere classified @ 50%
19 ENVIRONMENTAL POLICY, CLIMATE CHANGE AND NATURAL HAZARDS > 1999 Other environmental policy, climate change and natural hazards > 199999 Other environmental policy, climate change and natural hazards not elsewhere classified @ 50%
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