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Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions

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  • Masayuki Uchida

Abstract

. We consider an asymptotically efficient estimator of the drift parameter for a multi‐dimensional diffusion process with small dispersion parameter ɛ. In the situation where the sample path is observed at equidistant times k/n, k = 0, 1, …, n, we study asymptotic properties of an M‐estimator derived from an approximate martingale estimating function as ɛ tends to 0 and n tends to ∞ simultaneously.

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  • Masayuki Uchida, 2004. "Estimation for Discretely Observed Small Diffusions Based on Approximate Martingale Estimating Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(4), pages 553-566, December.
    • Handle: RePEc:bla:scjsta:v:31:y:2004:i:4:p:553-566
    DOI: 10.1111/j.1467-9469.2004.00406.x
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    1. Yuji Sakamoto & Nakahiro Yoshida, 2004. "Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(3), pages 545-597, September.
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    Cited by:

    1. Ma, Chunhua & Yang, Xu, 2014. "Small noise fluctuations of the CIR model driven by α-stable noises," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 1-11.
    2. Yang, Xu, 2017. "Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 18-27.
    3. Yusuke Kaino & Masayuki Uchida, 2018. "Hybrid estimators for small diffusion processes based on reduced data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(7), pages 745-773, October.
    4. Jakobsen, Nina Munkholt & Sørensen, Michael, 2019. "Estimating functions for jump–diffusions," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3282-3318.
    5. Long, Hongwei & Shimizu, Yasutaka & Sun, Wei, 2013. "Least squares estimators for discretely observed stochastic processes driven by small Lévy noises," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 422-439.
    6. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    7. Guy, Romain & Larédo, Catherine & Vergu, Elisabeta, 2014. "Parametric inference for discretely observed multidimensional diffusions with small diffusion coefficient," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 51-80.
    8. Ogihara, Teppei & Yoshida, Nakahiro, 2014. "Quasi-likelihood analysis for nonsynchronously observed diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 124(9), pages 2954-3008.
    9. Tetsuya Kawai & Masayuki Uchida, 2023. "Adaptive inference for small diffusion processes based on sampled data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(6), pages 643-696, August.
    10. Ren, Panpan & Wu, Jiang-Lun, 2021. "Least squares estimation for path-distribution dependent stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    11. Long, Hongwei, 2009. "Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2076-2085, October.
    12. Friedrich Hubalek & Petra Posedel, 2008. "Asymptotic analysis for a simple explicit estimator in Barndorff-Nielsen and Shephard stochastic volatility models," Papers 0807.3479, arXiv.org.
    13. Nina Munkholt Jakobsen & Michael Sørensen, 2015. "Efficient Estimation for Diffusions Sampled at High Frequency Over a Fixed Time Interval," CREATES Research Papers 2015-33, Department of Economics and Business Economics, Aarhus University.
    14. Uchida, Masayuki, 2008. "Approximate martingale estimating functions for stochastic differential equations with small noises," Stochastic Processes and their Applications, Elsevier, vol. 118(9), pages 1706-1721, September.
    15. Uchida, Masayuki & Yoshida, Nakahiro, 2013. "Quasi likelihood analysis of volatility and nondegeneracy of statistical random field," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2851-2876.
    16. Gloter, Arnaud & Sørensen, Michael, 2009. "Estimation for stochastic differential equations with a small diffusion coefficient," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 679-699, March.
    17. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    18. Friedrich Hubalek & Petra Posedel, 2011. "Joint analysis and estimation of stock prices and trading volume in Barndorff-Nielsen and Shephard stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 11(6), pages 917-932.
    19. Yasutaka Shimizu, 2017. "Threshold Estimation for Stochastic Processes with Small Noise," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 951-988, December.
    20. Héctor Araya & Soledad Torres & Ciprian A. Tudor, 2024. "Least squares estimation for the Ornstein–Uhlenbeck process with small Hermite noise," Statistical Papers, Springer, vol. 65(7), pages 4745-4766, September.

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