Minimum 𝐿₁-norm estimation for fractional Ornstein–Uhlenbeck type process

Prakasa Rao, B.

doi:10.1090/S0094-9000-05-00657-5

Minimum $L_1$-norm estimation for fractional Ornstein–Uhlenbeck type process

Author: B. L. S. Prakasa Rao
Journal: Theor. Probability and Math. Statist. 71 (2005), 181-189
MSC (2000): Primary 62M09; Secondary 60G15
DOI: https://doi.org/10.1090/S0094-9000-05-00657-5
Published electronically: December 28, 2005
MathSciNet review: 2144330
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the asymptotic properties of the minimum $L_1$-norm estimator of the drift parameter for fractional Ornstein–Uhlenbeck type process satisfying a linear stochastic differential equation driven by a fractional Brownian motion.

References

Elisa Alòs, Olivier Mazet, and David Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), no. 2, 766–801. MR 1849177, DOI https://doi.org/10.1214/aop/1008956692
Patrick Cheridito, Hideyuki Kawaguchi, and Makoto Maejima, Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab. 8 (2003), no. 3, 14. MR 1961165, DOI https://doi.org/10.1214/EJP.v8-125
Paul Doukhan, George Oppenheim, and Murad S. Taqqu (eds.), Theory and applications of long-range dependence, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1956041
Gustaf Gripenberg and Ilkka Norros, On the prediction of fractional Brownian motion, J. Appl. Probab. 33 (1996), no. 2, 400–410. MR 1385349, DOI https://doi.org/10.2307/3215063
Marc Henry and Paolo Zaffaroni, The long-range dependence paradigm for macroeconomics and finance, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 417–438. MR 1957502

H. E. Hurst, Long term storage capacity of reservoirs (with discussion), Trans. Am. Soc. Civ. Eng. 116 (1951), 770–808.

M. L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Stat. Inference Stoch. Process. 5 (2002), no. 3, 229–248. MR 1943832, DOI https://doi.org/10.1023/A%3A1021220818545

M. L. Kleptsyna, A. Le Breton, and M.-C. Roubaud, Parameter estimation and optimal filtering for fractional type stochastic systems, Stat. Inference Stoch. Process. 3 (2000), no. 1-2, 173–182. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). MR 1819294, DOI https://doi.org/10.1023/A%3A1009923431187

Y. Kutoyants and P. Pilibossian, On minimum $L_1$-norm estimate of the parameter of the Ornstein-Uhlenbeck process, Statist. Probab. Lett. 20 (1994), no. 2, 117–123. MR 1293288, DOI https://doi.org/10.1016/0167-7152%2894%2990026-4

Alain Le Breton, Filtering and parameter estimation in a simple linear system driven by a fractional Brownian motion, Statist. Probab. Lett. 38 (1998), no. 3, 263–274. MR 1629915, DOI https://doi.org/10.1016/S0167-7152%2898%2900029-7

P. W. Millar, A general approach to the optimality of minimum distance estimators, Trans. Amer. Math. Soc. 286 (1984), no. 1, 377–418. MR 756045, DOI https://doi.org/10.1090/S0002-9947-1984-0756045-0

Alberto Montanari, Long-range dependence in hydrology, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 461–472. MR 1957504

Ilkka Norros, Esko Valkeila, and Jorma Virtamo, An elementary approach to a Girsanov formula and other analytical results on fractional Brownian motions, Bernoulli 5 (1999), no. 4, 571–587. MR 1704556, DOI https://doi.org/10.2307/3318691

Ilkka Norros, Large deviations of queues with long-range dependent input, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 409–415. MR 1957501

A. A. Novikov, Sequential estimation of the parameters of processes of diffusion type, Mat. Zametki 12 (1972), 627–638 (Russian). MR 317493

Alexander Novikov and Esko Valkeila, On some maximal inequalities for fractional Brownian motions, Statist. Probab. Lett. 44 (1999), no. 1, 47–54. MR 1706311, DOI https://doi.org/10.1016/S0167-7152%2898%2900290-9

Vladas Pipiras and Murad S. Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields 118 (2000), no. 2, 251–291. MR 1790083, DOI https://doi.org/10.1007/s440-000-8016-7

B. L. S. Prakasa Rao, Asymptotic theory of statistical inference, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987. MR 874342

B. L. S. Prakasa Rao, Statistical inference for diffusion type processes, Kendall’s Library of Statistics, vol. 8, Edward Arnold, London; Oxford University Press, New York, 1999. MR 1717690

B. L. S. Prakasa Rao, Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion, Random Oper. Stochastic Equations 11 (2003), no. 3, 229–242. MR 2009183, DOI https://doi.org/10.1163/156939703771378581

B. L. S. Prakasa Rao, Berry–Esseen bound for MLE for linear stochastic differential equations driven by fractional Brownian motion, Preprint, Indian Statistical Institute, New Delhi, 2003b.

B. L. S. Prakasa Rao, Sequential estimation for fractional Ornstein-Uhlenbeck type process, Sequential Anal. 23 (2004), no. 1, 33–44. MR 2066971, DOI https://doi.org/10.1081/SQA-120030193

Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689

Murad S. Taqqu, Fractional Brownian motion and long-range dependence, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 5–38. MR 1956042

Walter Willinger, Vern Paxson, Rolf H. Riedi, and Murad S. Taqqu, Long-range dependence and data network traffic, Theory and applications of long-range dependence, Birkhäuser Boston, Boston, MA, 2003, pp. 373–407. MR 1957500