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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

 

Estimation of a matrix-valued parameter of an autoregressive process with nonstationary noise


Authors: A. P. Yurachkivskii and D. O. Ivanenko
Translated by: V. Zayats
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 72 (2005).
Journal: Theor. Probability and Math. Statist. 72 (2006), 177-191
MSC (2000): Primary 62F12; Secondary 60F05
Published electronically: September 6, 2006
MathSciNet review: 2168147
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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that $ \check{A}_n$ is the least squares estimator constructed from $ n$ observations of an unknown matrix $ A$ in an autoregressive process $ \xi_{k}=A\xi_{k-1}+\varepsilon_{k}$. Under the assumption that the sequence $ (\varepsilon_k)$ is a martingale difference, not necessarily stationary and ergodic, we find the limit distribution as $ n\to\infty$ of the statistic $ \sqrt{n}(\check{A}_{n}-A)$ by using methods of stochastic analysis. This limit distribution may be different from the normal distribution.


References [Enhancements On Off] (What's this?)

  • 1. M. Arato, Lineinye stokhasticheskie sistemy s postoyannymi koeffitsientami, “Nauka”, Moscow, 1989 (Russian). Statisticheskii podkhod. [A statistical approach]; Translated from the English by V. K. Malinovskiĭ\ and V. I. Khokhlov; Translation edited and with a preface by Yu.\ A. Rozanov. MR 1011456 (90i:62108)
  • 2. V. O. Koval, Limit theorems for operator-normalized random vectors. II, Teor. Imov{\={\i\/}}\kern.15emr. Mat. Stat. 62 (2000), 37-47; English transl. in Theory Probab. Math. Statist. 62 (2001), 39-49. MR 1871507 (2002j:60051
  • 3. R. Sh. Liptser and A. N. Shiryayev, Theory of martingales, Mathematics and its Applications (Soviet Series), vol. 49, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by K. Dzjaparidze [Kacha Dzhaparidze]. MR 1022664 (90j:60046)
  • 4. Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 1987. MR 959133 (89k:60044)
  • 5. T. W. Anderson, The statistical analysis of time series, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0283939 (44 #1169)
  • 6. I. Vuchkov, L. Boyadzhieva, and E. Solakov, Prikladnoi lineinyi regressionnyi analiz, \cyr Bibliotechka Inostrannykh Knig dlya Èkonomistov i Statistikov. [A Library of Foreign Books for Economists and Statisticians], “Finansy i Statistika”, Moscow, 1987 (Russian). With the collaboration of D. Boyadzhiev; Translated from the Bulgarian and with a preface by Yu.\ P. Adler. MR 935130 (89f:62058)
  • 7. A. Ya. Dorogovtsev, Teoriya otsenok parametrov sluchainykh protsessov, “Vishcha Shkola”, Kiev, 1982 (Russian). MR 668517 (84h:62122)

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Additional Information

A. P. Yurachkivskii
Affiliation: Department of Mathematics and Theoretical Radiophysics, Faculty of Radiophysics, Taras Shevchenko National University, Glushkov Ave. 2, Building 5, 03127 Kyïv, Ukraine
Email: yap@univ.kiev.ua

D. O. Ivanenko
Affiliation: Department of Mathematics and Theoretical Radiophysics, Faculty of Radiophysics, Taras Shevchenko National University, Glushkov Ave. 2, Building 5, 03127 Kyïv, Ukraine
Email: ida@univ.kiev.ua

DOI: http://dx.doi.org/10.1090/S0094-9000-06-00675-2
PII: S 0094-9000(06)00675-2
Received by editor(s): May 24, 2004
Published electronically: September 6, 2006
Article copyright: © Copyright 2006 American Mathematical Society