Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-18T10:01:35.815Z Has data issue: false hasContentIssue false
  • English
  • Français

On the estimation of a harmonic component in a time series with stationary dependent residuals

Published online by Cambridge University Press:  01 July 2016

A. M. Walker
A. M. Walker*
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

Abstract

Let observations (X1, X2, …, Xn) be obtained from a time series {Xt} such that where the ɛt are independently and identically distributed random variables each having mean zero and finite variance, and the gu(θ) are specified functions of a vector-valued parameter θ. This paper presents a rigorous derivation of the asymptotic distributions of the estimators of A, B, ω and θ obtained by an approximate least-squares method due to Whittle (1952). It is a sequel to a previous paper (Walker (1971)) in which a similar derivation was given for the special case of independent residuals where gu(θ) = 0 for u > 0, the parameter θ thus being absent.

Keywords

TIME SERIESHARMONIC COMPONENTSLINEAR RESIDUAL PROCESSLEAST SQUARES ESTIMATIONASYMPTOTIC DISTRIBUTION OF ESTIMATES
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 2 , August 1973 , pp. 217 - 241
Copyright
Copyright © Applied Probability Trust 1973 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Hannan, E. J. (1971) Non-linear time series regression. J. Appl. Prob. 8, 767–780.Google Scholar
Olshen, R. A. (1967) Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Prob. 4, 508–528.Google Scholar
Walker, A. M. (1964) Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time series. J. Austral. Math. Soc. 4, 363–384.Google Scholar
Walker, A. M. (1965) Some asymptotic results for the periodogram of a stationary time series. J. Austral. Math. Soc. 5, 107–128.Google Scholar
Walker, A. M. (1969) On the estimation of a harmonic component in a time series with stationary residuals. Bull. Inst. Internat. Statist. 42, II, 374–376.Google Scholar
Walker, A. M. (1970) On the estimation of a harmonic component in a time series with stationary residuals; II, dependent residuals. Technical Report No. 50, Department of Statistics, Stanford University.Google Scholar
Walker, A. M. (1971) On the estimation of a harmonic component in a time series with stationary independent residuals. Biometrika 58, 21–36.Google Scholar
Whittle, P. (1952) The simultaneous estimation of a time series' harmonic components and covariance structure. Trabajos Estadist. 3, 43–57.Google Scholar