Robust estimation for continuous-time linear models with memory
Authors:
Mamikon S. Ginovyan and Artur A. Sahakyan
Journal:
Theor. Probability and Math. Statist. 95 (2017), 81-98
MSC (2010):
Primary 60F05, 60G22; Secondary 62G05, 62G20
DOI:
https://doi.org/10.1090/tpms/1023
Published electronically:
February 28, 2018
MathSciNet review:
3631645
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References |
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Additional Information
Abstract: In time series analysis, much of the statistical inferences about unknown spectral parameters or spectral functionals are concerned with the discrete-time stationary models, in which case it is assumed that the models are centered, or have constant means. The present paper deals with a question involving robustness of inferences, carried out on Lévy-driven continuous-time linear models, possibly exhibiting long memory, contaminated by a small trend. We show that a smoothed periodogram approach to both parametric and nonparametric estimation is robust to the presence of a small trend in the model.
References
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References
- V. V. Anh, J. M. Angulo, and M. D. Ruiz-Medina, Possible long-range dependence in fractional random fields, J. Statist. Planning Inference 80 (1999), 95–110. MR 1713795
- V. V. Anh, N. N. Leonenko, and R. McVinish, Models for fractional Riesz–Bessel motion and related processes, Fractals 9 (2001), 329–346.
- V. V. Anh, N. N. Leonenko, and L. Sakhno, On a class of minimum contrast estimators for fractional stochastic processes and fields, J. Statist. Planning Inference 123 (2004), 161–185. MR 2058127
- V. V. Anh, N. N. Leonenko, and L. Sakhno, Minimum contrast estimation of random processes based on information of second and third orders, J. Statist. Planning Inference 137 (2007), 1302–1331. MR 2301481
- F. Avram, N. N. Leonenko, and L. 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: Probability and Statistics 14 (2010), 210–255. MR 2741966
- S. Bai, M. S. Ginovyan, and M. S. Taqqu, Limit theorems for quadratic forms of Levy-driven continuous-time linear processes, Stochast. Process. Appl. 126 (2016), 1036–1065. MR 3461190
- J. Beran, Y. Feng, S. Ghosh, and R. Kulik, Long-Memory Processes. Probabilistic Properties and Statistical Methods, Springer, New York, 2013. MR 3075595
- P. J. Brockwell, Recent results in the theory and applications of CARMA processes, Annals of the Institute of Statistical Mathematics 66 (2014), no. 4, 647–685. MR 3224604
- I. Casas and J. Gao, Econometric estimation in long-range dependent volatility models: Theory and practice, Journal of Econometrics 147 (2008), 72–83. MR 2472982
- M. J. Chambers, The Estimation of Continuous Parameter Long-Memory Time Series Models, Econometric Theory 12 (1996), no. 2, 374–390. MR 1395038
- R. Dahlhaus, Efficient parameter estimation for self-similar processes, Ann. Statist. 17 (1989), 1749–1766. MR 1026311
- R. Dahlhaus and W. Wefelmeyer, Asymptotically optimal estimation in misspecified time series models, Ann. Statist. 24 (1996), 952–974. MR 1401832
- K. Dzhaparidze, Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series, Springer, New York, 1986. MR 812272
- R. Fox and M. S. Taqqu, Large-sample properties of parameter estimation for strongly dependent stationary Gaussian time series, Ann. Statist. 14 (1986), 517–532. MR 840512
- J. Gao, Modelling long-range dependent Gaussian processes with application in continuous-time financial models, J. Appl. Probab. 41 (2004), 467–482. MR 2052585
- J. Gao, V. V. Anh, C. Heyde, and Q. Tieng, Parameter Estimation of Stochastic Processes with Long-range Dependence and Intermittency, J. Time Ser. Anal. 22 (2001), 517–535. MR 1859563
- J. Gao, V. V. Anh, and C. Heyde, Statistical estimation of nonstationary Gaussian process with long-range dependence and intermittency, Stochast. Process. Appl. 99 (2002), 295–321. MR 1901156
- M. S. Ginovyan, Asymptotically efficient nonparametric estimation of functionals of a spectral density having zeros, Theory Probab. Appl. 33 (1988), no. 2, 296–303. MR 954578
- M. S. Ginovyan, On estimating the value of a linear functional of the spectral density of a Gaussian stationary process, Theory Probab. Appl. 33 (1988), no. 4, 722–726. MR 979749
- M. S. Ginovyan, On Toeplitz type quadratic functionals in Gaussian stationary process, Probab. Theory Relat. Fields 100 (1994), 395–406. MR 1305588
- M. S. Ginovyan, Asymptotic properties of spectrum estimate of stationary Gaussian processes, J. Cont. Math. Anal. 30 (1995), no. 1, 1–16. MR 1643528
- M. S. Ginovyan, Asymptotically efficient nonparametric estimation of nonlinear spectral functionals, Acta Appl. Math. 78 (2003), 145–154. MR 2024019
- M. S. Ginovyan, Efficient Estimation of Spectral Functionals for Gaussian Stationary Models, Comm. Stochast. Anal. 5 (2011), no. 1, 211–232. MR 2808543
- M. S. Ginovyan, Efficient Estimation of Spectral Functionals for Continuous-time Stationary Models, Acta Appl. Math. 115 (2011), no. 2, 233–254. MR 2818916
- M. S. Ginovyan and A. A. Sahakyan, Limit Theorems for Toeplitz quadratic functionals of continuous-time stationary process, Probab. Theory Relat. Fields 138 (2007), 551–579. MR 2299719
- L. Giraitis and D. Surgailis, A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate, Probab. Theory Relat. Fields 86 (1990), 87–104. MR 1061950
- L. Giraitis, H. L. Koul, and D. Surgailis, Large Sample Inference for Long Memory Processes, Imperial College Press, London, 2012. MR 2977317
- R. Z. Has’minskii and I. A. Ibragimov, Asymptotically efficient nonparametric estimation of functionals of a spectral density function, Probab. Theory Related Fields 73 (1986), 447–461. MR 859842
- C. Heyde and W. Dai, On the robustness to small trends of estimation based on the smoothed periodogram, J. Time Ser. Anal. 17 (1996), no. 2, 141–150. MR 1381169
- I. A. Ibragimov and R. Z. Khasminskii, Asymptotically normal families of distributions and efficient estimation, Ann. Statist. 19 (1991), 1681–1724. MR 1135145
- A. V. Ivanov and V. V. Prikhod’ko, On the Whittle Estimator of the Parameters of Spectral Density of Random Noise in the Nonlinear Regression Model, Ukrainian Math. J. 67 (2016), no. 8, 1183–1203. MR 3473712
- A. V. Ivanov and V. V. Prikhod’ko, Asymptotic properties of Ibragimov’s estimator for a parameter of the spectral density of the random noise in a nonlinear regression model, Teor. Imovir. ta Matem. Statyst. 93 (2015), 50–66.
- H. L. Koul and D. Surgailis, Asymptotic normality of the Whittle estimator in linear regression model with long memory errors, Statist. Inference for Stochast. Processes 3 (2000), no. 1, 129–147. MR 1819291
- N. N. Leonenko and L. Sakhno, On the Whittle estimators for some classes of continuous-parameter random processes and fields, Stat & Probab. Letters 76 (2006), 781–795. MR 2266092
- P. W. Millar, Non-parametric applications of an infinite dimensional convolution theorem, Z. Wahrsch. verw. Gebiete 68 (1985), 545–556. MR 772198
- V. Solo, Intrinsic random functions and the paradox of $1/f$ noise, SIAM J. Appl. Math. 52 (1992), 270–291. MR 1148328
- M. Taniguchi, Minimum contrast estimation for spectral densities of stationary processes, J. Roy. Statist. Soc., Ser. B 49 (1987), no. 3, 315–325. MR 928940
- M. Taniguchi and Y. Kakizawa, Asymptotic Theory of Statistical Inference for Time Series, Springer, New York, 2000. MR 1785484
- H. Tsai and K. S. Chan, Quasi-maximum likelihood estimation for a class of continuous-time long memory processes, J. Time Ser. Anal. 26 (2005), no. 5, 691–713. MR 2188305
- P. Whittle, Hypothesis Testing in Time Series, Hafner, New York, 1951. MR 0040634
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Additional Information
Mamikon S. Ginovyan
Affiliation:
Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215
Email:
ginovyan@math.bu.edu
Artur A. Sahakyan
Affiliation:
Department of Mathematics and Mechanics, Yerevan State University, 1 Alex Manoogian, Yerevan, 0025, Armenia
Email:
sart@ysu.am
Keywords:
Trend,
robust inference,
Lévy-driven continuous-time model,
memory,
smoothed periodogram,
parametric and nonparametric estimation
Received by editor(s):
October 17, 2016
Published electronically:
February 28, 2018
Additional Notes:
The research of the first author was partially supported by National Science Foundation Grant #DMS-1309009 at Boston University.
Article copyright:
© Copyright 2018
American Mathematical Society
