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Asymptotic properties of least-squares estimates of parameters of the spectrum of a stationary non-deterministic time-series

Published online by Cambridge University Press:  09 April 2009

A. M. Walker
A. M. Walker
Affiliation:
University of Cambridge and Institue of Advanced Studies, Australian National University.
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Let {xt} (t = 0, ±1, ±2 …) be a stationary non-deterministic time series with E(x2t) < ∞, E(xt) = 0, and let its spectrum be continuous (strictly absolutely continuous) so that the spectral distribution function is the spectral density function. It is well known that {xt} then has a unique one-sided movingaverage representation where .

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 4 , Issue 3 , August 1964 , pp. 363 - 384
Copyright
Copyright © Australian Mathematical Society 1964

References

[1]Anderson, T. W., On asymptotic distributions of estimates of parameters of stochastic difference equations. Ann. Math. Statist. 30 (1959), 676–87.CrossRefGoogle Scholar
[2]Nartlett, M. S., An Introduction to Stochastic Processes. (1955), Cambridge University Press.Google Scholar
[3]Cramér, H., Mathematical Methods of Statistics. (1946), Princeton.Google Scholar
[4]Grenender, U., On the estimation of regression coefficients in the case of an autocorrelated disturbance. Ann. Math. Statist. 25 (1954), 252–72.CrossRefGoogle Scholar
[5]Grenander, U. and Rosenblatt, M., Statistical spectural analysis of time series arising from stationary stochastic Processes. Ann. Math. Statist. 24 (1953), 573–58.CrossRefGoogle Scholar
[6]Grenander, U. and Rosenblatt, M., Statistical Analysis of Stationary Time Series. (1957), Wiley, New York.CrossRefGoogle Scholar
[7]Hannan, E. J., Time Series Analysis. (1960), Methuen.Google Scholar
[8]Silvey, S. D., Anote on maximum- likelihood in the case of dependent random variables. J. Roy. Statist. Soc. B. 23 (1961), 444–52.Google Scholar
[9]Titchmarch, E. C., Theory of Functions, 2nd edn. (1944), Oxford University Press.Google Scholar
[10]Wald, A., Note on the consistency of the maximum-likelihood estimate. Ann. Math. Statist. 20 (1949), 595–601.CrossRefGoogle Scholar
[11]Walker, A. M., The asympotic distribution of serial correlation coefficients for autoregressive processes with dependent residuals. Proc. Cambridge Philos. Soc. 50 (1954), 60–64.CrossRefGoogle Scholar
[12]Whittle, P., Hypothesis Testing in Time series Analysis. (1951), Uppsala: Almqvist and Wiksell.Google Scholar
[13]Whittle, P., Some results in time series analysis. Skand. Aktuar. 35 (1952), 48–60.Google Scholar
[14]Whittle, P., Estimation and information in stationary time series. Ark. Mat. Astr. Fys 2 (1953), 423–34.CrossRefGoogle Scholar
[15]Whittle, P., A study in the analusis of Stationary Times Series, Appendix 2. (1954) Uppsala :Almqvist and Wiksell.Google Scholar
[16]Wolfowitz, J., On wald's proff of the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 (1949), 601–2.CrossRefGoogle Scholar
[17]Mann, H. B. and Wald, A., On the statistical treatment of linear stochastic difference equations. Econometrica 11 (1943), 173–220.CrossRefGoogle Scholar