Theory of Probability and Mathematical Statistics

Author(s): B. L. S. Prakasa Rao
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 71 (2004).
Journal: Theor. Probability and Math. Statist. No. 71 (2005), 181-189.
MSC (2000): Primary 62M09; Secondary 60G15
Posted: December 28, 2005
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Abstract: We investigate the asymptotic properties of the minimum

-norm estimator of the drift parameter for fractional Ornstein-Uhlenbeck type process satisfying a linear stochastic differential equation driven by a fractional Brownian motion.

1.: E. Alos, O. Mazet, and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), 766-801. MR 1849177 (2002g:60083)
2.: P. Cheridito, H. Kawaguchi and M. Maejima, Fractional Ornstein-Uhlenbeck processes, Electronic Journal of Probability 8 (2003), 1-14. MR 1961165 (2003m:60096)
3.: P. Doukhan, G. Oppenheim, and M. S. Taqqu, Theory of Long-Range Dependence, Birkhäuser, Boston, 2003. MR 1956041 (2003h:60004)
4.: G. Gripenberg and I. Norros, On the prediction of fractional Brownian motion, J. Appl. Probab. 33 (1996), 400-410. MR 1385349 (97a:60058)
5.: M. Henry and P. Zafforoni, The long-range dependence paradigm for macroeconomics and finance, Theory of Long-Range Dependence (P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds.), Birkhäuser, Boston, 2003, pp. 417-438. MR 1957502
6.: H. E. Hurst, Long term storage capacity of reservoirs (with discussion), Trans. Am. Soc. Civ. Eng. 116 (1951), 770-808.
7.: M. L. Kleptsyna and A. Le Breton, Statistical analysis of the fractional Ornstein-Uhlenbeck type process, Statist. Infer. Stoch. Proces. 5 (2002), 229-248. MR 1943832 (2003i:62140)
8.: M. L. Kleptsyna, A. Le Breton, M.-C. Roubaud, Parameter estimation and optimal filtering for fractional type stochastic systems, Statist. Infer. Stoch. Proces. 3 (2000), 173-182. MR 1819294 (2002c:60089)
9.: Yu. Kutoyants and P. Pilibossian, On minimum -norm estimate of the parameter of the Ornstein-Uhlenbeck process, Statist. Probab. Lett. 20 (1994), 117-123. MR 1293288 (95g:62161)
10.: A. Le Breton, Filtering and parameter estimation in a simple linear model driven by a fractional Brownian motion, Statist. Probab. Lett. 38 (1998), 263-274. MR 1629915 (99c:60088)
11.: P. W. Millar, A general approach to the optimality of the minimum distance estimators, Trans. Amer. Math. Soc. 286 (1984), 377-418. MR 0756045 (85m:62057)
12.: A. Montanari, Long-range dependence in hydrology, Theory of Long-Range Dependence (P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds.), Birkhäuser, Boston, 2003, pp. 461-472. MR 1957504
13.: I. Norros, E. Valkeila, and J. Virtamo, An elementary approach to a Girsanov type formula and other analytical results on fractional Brownian motion, Bernoulli 5 (1999), 571-587. MR 1704556 (2000f:60053)
14.: I. Norros, Large deviations of queues with long-range dependent input, Theory of Long-Range Dependence (P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds.), Birkhäuser, Boston, 2003, pp. 409-415. MR 1957501
15.: A. A. Novikov, Sequential estimation of the parameters of diffusion processes, Mathematical Notes 12 (1972), 812-818. MR 0317493 (47:6040)
16.: A. A. Novikov and E. Valkeila, On some maximal inequalities for fractional Brownian motion, Statist. Probab. Lett. 44 (1999), 47-54. MR 1706311 (2000g:60076)
17.: V. Papiras and M. Taqqu, Integration questions related to fractional Brownian motion, Prob. Theory and Rel. Fields 118 (2000), 121-291. MR 1790083 (2002c:60091)
18.: B. L. S. Prakasa Rao, Asymptotic Theory of Statistical Inference, Wiley, New York, 1987. MR 0874342 (88b:62001)
19.: B. L. S. Prakasa Rao, Statistical Inference for Diffusion Type Processes, Arnold, London and Oxford University Press, New York, 1999. MR 1717690 (2000k:62149)
20.: B. L. S. Prakasa Rao, Parametric estimation for linear stochastic differential equations driven by fractional Brownian motion, Random Oper. Stoch. Eq. 11 (2003), 229-242. MR 2009183 (2005a:62188)
21.: 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.
22.: B. L. S. Prakasa Rao, Sequential estimation for fractional Ornstein-Uhlenbeck type process, Sequential Analysis 23 (2004), 33-44. MR 2066971 (2005d:62127)
23.: S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science, 1993. MR 1347689 (96d:26012)
24.: M. Taqqu, Fractional Brownian motion and long-range dependence, Theory of Long-Range Dependence (P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds.), Birkhäuser, Boston, 2003, pp. 5-38. MR 1956042
25.: W. Willinger, V. Paxson, R. H. Riedi, and M. Taqqu, Long-range dependence and data network traffic, Theory of Long-Range Dependence (P. Doukhan, G. Oppenheim, and M. S. Taqqu, eds.), Birkhäuser, Boston, 2003, pp. 373-407. MR 1957500

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 62M09, 60G15

B. L. S. Prakasa Rao
Affiliation: Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 500 046, India
Email: blsprsm@uohyd.ernet.in

DOI: 10.1090/S0094-9000-05-00657-5
PII: S 0094-9000(05)00657-5
Keywords: Minimum $L_1$-norm estimation, fractional Ornstein--Uhlenbeck type process, fractional Brownian motion
Received by editor(s): 25/NOV/2003
Posted: December 28, 2005
Copyright of article: Copyright 2005, American Mathematical Society

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