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Asymptotic properties of the corrected score estimator in the autoregressive model with measurement errors


Authors: D. S. Pupashenko, S. V. Shklyar and A. G. Kukush
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 89 (2013).
Journal: Theor. Probability and Math. Statist. 89 (2014), 169-180
MSC (2010): Primary 62F12; Secondary 62F10
DOI: https://doi.org/10.1090/S0094-9000-2015-00943-1
Published electronically: January 26, 2015
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Abstract: The autoregressive model with errors in variables and with a normally distributed control sequence is considered. For the main sequence, two cases are considered: (a) the main sequence has a stationary distribution, and (b) the initial distribution is arbitrary, independent of the control sequence, and has a finite fourth moment. Here the elements of the main sequence are not observed directly, but surrogate data that include a normally distributed additive error are observed. The errors and main sequence are assumed to be mutually independent.

We estimate the unknown parameter using the Corrected Score method and in both cases prove the strict consistency and asymptotic normality of the estimator. To prove the asymptotic normality we apply the theory of strong mixing sequences. Finally, we compare the efficiency of the Least Squares (naive) estimator and the Corrected Score estimator in the forecasting problem and conclude that the naive estimator gives a better forecast.


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Additional Information

D. S. Pupashenko
Affiliation: Taras Shevchenko National University of Kyiv, Volodymyrska st. 64, 01601, Kyiv, Ukraine
Email: daria.pupashenko@gmail.com

S. V. Shklyar
Affiliation: Taras Shevchenko National University of Kyiv, Volodymyrska st. 64, 01601, Kyiv, Ukraine
Email: shklyar@mail.univ.kiev.ua

A. G. Kukush
Affiliation: Taras Shevchenko National University of Kyiv, Volodymyrska st. 64, 01601, Kyiv, Ukraine
Email: alexander_kukush@univ.kiev.ua

DOI: https://doi.org/10.1090/S0094-9000-2015-00943-1
Keywords: Autoregressive model, measurement errors, stationary process, strong mixing sequences, least squares estimation, corrected score estimation, efficiency comparison
Received by editor(s): April 1, 2013
Published electronically: January 26, 2015
Article copyright: © Copyright 2015 American Mathematical Society