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Asymptotic properties of time domain gaussian estimators

Published online by Cambridge University Press:  01 July 2016

R. Kohn
R. Kohn*
Affiliation:
The Australian National University, Canberra
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Abstract

We consider a general non-linear multivariate time series model which can be parameterized by a finite and fixed number of parameters and which can be rewritten, if necessary, in a form such that the disturbances are stationary martingale differences. Given a series of discrete, equally spaced observations we prove the strong consistency and asymptotic normality of the Gaussian estimators of the parameters, the parameters possibly being subject to non-linear constraints. Because the normal equations are usually highly non-linear it may be difficult to obtain explicit expressions for the Gaussian estimates. To overcome this problem we use a Gauss–Newton type algorithm to obtain a sequence of iterates which converge to, and have the same asymptotic properties as, the Gaussian estimates.

Keywords

CENTRAL LIMIT THEOREMCONSTRAINTSGAUSS-NEWTONGAUSSIAN ESTIMATESMARTINGALE DIFFERENCESMULTIVARIATE TIME SERIESNON-LINEAR
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 2 , June 1978 , pp. 339 - 359
Copyright
Copyright © Applied Probability Trust 1978 

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References

Aitchison, J. and Silvey, S. D. (1958) Maximum-likelihood estimation of parameters subject to restraints. Ann. Math. Statist. 29, 813–828.Google Scholar
Anderson, T. W. (1971) The Statistical Analysis of Time Series. Wiley, New York.Google Scholar
Bard, Y. (1974) Nonlinear Parameter Estimation. Academic Press, New York.Google Scholar
Brown, B. M. and Eagleson, G. K. (1971) Martingale convergence to infinitely divisible laws with finite variances. Trans. Amer. Math. Soc. 16, 449–453.Google Scholar
Fisher, F. M. (1966) The Identification Problem in Econometrics. McGraw-Hill, New York.Google Scholar
Hannan, E. J. (1971) Non-linear time series regression. J. Appl. Prob. 8, 767–780.Google Scholar
Hannan, E. J. and Heyde, C. C. (1972) On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 43, 2058–2066.Google Scholar
Heyde, C. C. (1973) An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes. J. Appl. Prob. 10, 146–157.Google Scholar
Jennrich, R. I. (1969) Asymptotic properties of non-linear least squares estimators. Ann. Math. Statist. 40, 633–643.Google Scholar
Kohn, R. (1977) Some asymptotic results for vector linear time series models. Submitted for publication.Google Scholar
Neveu, J. (1975) Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
Nicholls, D. F. (1976) The efficient estimation of vector linear time series models. Biometrika 63, 381–390.Google Scholar
Petrovski, I. G. (1966) Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, N. J.Google Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications, 2nd edn. Wiley, New York.Google Scholar
Robinson, P. M. (1972) Non-linear regression for multiple time series. J. Appl. Prob. 9, 758–768.Google Scholar
Rudin, W. (1964) Principles of Mathematical Analysis, 2nd edn. McGraw-Hill, New York.Google Scholar
Stout, W. F. (1974) Almost Sure Convergence. Academic Press, New York.Google Scholar
Whittle, P. (1962) Gaussian estimation in stationary time series. Bull. Internal Statist. Inst. 39, 105–129.Google Scholar