Parameter estimation for reciprocal gamma Ornstein–Uhlenbeck type processes
Authors:
N. Leonenko, L. Sakhno and N. Šuvak
Journal:
Theor. Probability and Math. Statist. 86 (2013), 137-154
MSC (2010):
Primary 60G10, 60J60, 62M05, 62M15
DOI:
https://doi.org/10.1090/S0094-9000-2013-00894-1
Published electronically:
August 20, 2013
MathSciNet review:
2986455
Full-text PDF Free Access
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Abstract: We consider parameter estimation for a process of Ornstein–Uhlenbeck type with reciprocal gamma marginal distribution, to be called reciprocal gamma Ornstein–Uhlenbeck (RGOU) process. We derive minimum contrast estimators of unknown parameters based on both the discrete and the continuous observations from the process as well as moments based estimators based on discrete observations. We prove that proposed estimators are consistent and asymptotically normal. The explicit forms of the asymptotic covariance matrices are determined by using the higher order spectral densities and cumulants of the RGOU process.
References
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References
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, The National Bureau of standards, 1964.
- V. V. Anh, C. C. Heyde, and N. N. Leonenko, Dynamic models of long-memory processes driven by Lévy noise, J. Appl. Probab. 39 (2002), no. 4, 730–747. MR 1938167 (2004c:60109)
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence, J. Appl. Probab. 41A (2004), 35–53. MR 2057564 (2005j:62169)
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, On a class of minimum contrast estimators for fractional stochastic processes and fields, J. Statist. Plann. Inference 123 (2004), 161–185. MR 2058127 (2005g:62046)
- V. V. Anh, N. N. Leonenko, and L. M. Sakhno, Minimum contrast estimation of random processes based on information of second and third orders, J. Statist. Plann. Inference 137 (2007), 1302–1331. MR 2301481 (2008h:62218)
- F. Avram, N. N. Leonenko, and L. M. Sakhno, On a Szegö type limit theorem, the Hölder–Young–Brascamp–Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields, ESAIM Probab. Stat. 14 (2010), 210–255. MR 2741966 (2011i:60041)
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- H. Hassani, Sum of the sample autocorrelation function, Random Oper. Stoch. Equ. 17 (2009), 125–130. MR 2560860 (2010i:62251)
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- C. C. Heyde and N. N. Leonenko, Student processes, Adv. in Appl. Probab. 37 (2005), 342–365. MR 2144557 (2005m:62161)
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- G. Jongbloed, F. H. Van der Meulen, and A. W. Van der Vaart, Nonparametric inference for Lévy-driven Ornstein–Uhlenbeck processes, Bernoulli 11 (2005), 759–791. MR 2172840 (2007a:62048)
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- N. N. Leonenko and L. M. Sakhno, On the Whittle estimators for some classes of continuous parameter random processes and fields, Stat. Probability Letters 76 (2006), 781–795. MR 2266092 (2009c:62123)
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- K. Spiliopoulos, Method of moments estimation of Ornstein–Uhlenbeck processes driven by general Lévy process, Ann. I.S.U.P. 53 (2009), 3–17. MR 2643269 (2011c:62275)
- S. Sun and X. Zhang, Empirical likelihood estimation of discretely sampled processes of OU type, Sci. China Ser. A 52 (2009), 908–931. MR 2504998 (2010f:62228)
- L. Valdivieso, W. Schoutens, and F. Tuerlinckx, Maximum likelihood estimation in processes of Ornstein–Uhlenbeck type, Stat. Inference Stoch. Process. 12 (2009), 1–19. MR 2486113 (2010i:62237)
- V. Witkovsky, Exact distribution of positive linear combinations of inverted chi-square random variables with odd degrees of freedom, Statist. Probab. Letters 56 (2002), 45–50. MR 1881529 (2002k:62033)
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Additional Information
N. Leonenko
Affiliation:
School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF244AG, UK
Email:
LeonenkoN@Cardiff.ac.uk
L. Sakhno
Affiliation:
Department of Probability, Statistics and Actuarial Mathematics, Mechanics and Mathematics Faculty, Taras Shevchenko National University of Kyiv, 64, Volodymyrs’ka St., 01601 Kyiv, Ukraine
Email:
lms@univ.kiev.ua
N. Šuvak
Affiliation:
Department of Mathematics, University of Osijek, Gajev Trg 6, HR-31 000 Osijek, Croatia
Email:
nsuvak@mathos.hr
Keywords:
Ornstein–Uhlenbeck type process,
reciprocal gamma distribution,
infinite divisibility,
self-decomposability,
parameter estimation,
method of moments,
minimum contrast method,
Ibragimov functionals,
Whittle functionals
Received by editor(s):
November 20, 2030
Published electronically:
August 20, 2013
Additional Notes:
Partly supported by the Commission of the European Communities grant PIRSES-GA-2008-230804 within the programme ‘Marie Curie Actions’ and the grant of the United Kingdom Association of Alumni and Friends of Croatian Universities (AMAC-UK)
Article copyright:
© Copyright 2013
American Mathematical Society