## The Annals of Statistics

### Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency

Hiroki Masuda

#### Abstract

This paper investigates the Gaussian quasi-likelihood estimation of an exponentially ergodic multidimensional Markov process, which is expressed as a solution to a Lévy driven stochastic differential equation whose coefficients are known except for the finite-dimensional parameters to be estimated, where the diffusion coefficient may be degenerate or even null. We suppose that the process is discretely observed under the rapidly increasing experimental design with step size $h_{n}$. By means of the polynomial-type large deviation inequality, convergence of the corresponding statistical random fields is derived in a mighty mode, which especially leads to the asymptotic normality at rate $\sqrt{nh_{n}}$ for all the target parameters, and also to the convergence of their moments. As our Gaussian quasi-likelihood solely looks at the local-mean and local-covariance structures, efficiency loss would be large in some instances. Nevertheless, it has the practically important advantages: first, the computation of estimates does not require any fine tuning, and hence it is straightforward; second, the estimation procedure can be adopted without full specification of the Lévy measure.

#### Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1593-1641.

Dates
First available in Project Euclid: 1 August 2013

https://projecteuclid.org/euclid.aos/1375362561

Digital Object Identifier
doi:10.1214/13-AOS1121

Mathematical Reviews number (MathSciNet)
MR3113823

Zentralblatt MATH identifier
1292.62124

Subjects
Primary: 62M05: Markov processes: estimation

#### Citation

Masuda, Hiroki. Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at high frequency. Ann. Statist. 41 (2013), no. 3, 1593--1641. doi:10.1214/13-AOS1121. https://projecteuclid.org/euclid.aos/1375362561

#### References

• [1] Aït-Sahalia, Y. and Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. Ann. Statist. 35 355–392.
• [2] Aït-Sahalia, Y. and Jacod, J. (2008). Fisher’s information for discretely sampled Lévy processes. Econometrica 76 727–761.
• [3] Bardet, J.-M. and Wintenberger, O. (2009). Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. Ann. Statist. 37 2730–2759.
• [4] Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 167–241.
• [5] Bhansali, R. J. and Papangelou, F. (1991). Convergence of moments of least squares estimators for the coefficients of an autoregressive process of unknown order. Ann. Statist. 19 1155–1162.
• [6] Chan, N. H. and Ing, C.-K. (2011). Uniform moment bounds of Fisher’s information with applications to time series. Ann. Statist. 39 1526–1550.
• [7] Dvoretzky, A. (1977). Asymptotic normality of sums of dependent random vectors. In Multivariate Analysis, IV (Proc. Fourth Internat. Sympos., Dayton, Ohio, 1975) 23–34. North-Holland, Amsterdam.
• [8] Findley, D. F. and Wei, C.-Z. (2002). AIC, overfitting principles, and the boundedness of moments of inverse matrices for vector autoregressions and related models. J. Multivariate Anal. 83 415–450.
• [9] Fort, G. and Roberts, G. O. (2005). Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab. 15 1565–1589.
• [10] Friedman, A. (2006). Stochastic Differential Equations and Applications. Dover Publications, Mineola, NY.
• [11] Gobet, E. (2002). LAN property for ergodic diffusions with discrete observations. Ann. Inst. Henri Poincaré Probab. Stat. 38 711–737.
• [12] Heyde, C. C. (1997). Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. Springer, New York.
• [13] Ibragimov, I. A. and Has’minskiĭ, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Applications of Mathematics 16. Springer, New York.
• [14] Inagaki, N. and Ogata, Y. (1975). The weak convergence of likelihood ratio random fields and its applications. Ann. Inst. Statist. Math. 27 391–419.
• [15] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [16] Jeganathan, P. (1982). On the convergence of moments of statistical estimators. Sankhyā Ser. A 44 213–232.
• [17] Jeganathan, P. (1989). A note on inequalities for probabilities of large deviations of estimators in nonlinear regression models. J. Multivariate Anal. 30 227–240.
• [18] Kessler, M. (1997). Estimation of an ergodic diffusion from discrete observations. Scand. J. Stat. 24 211–229.
• [19] Kulik, A. M. (2009). Exponential ergodicity of the solutions to SDE’s with a jump noise. Stochastic Process. Appl. 119 602–632.
• [20] Kulik, A. M. (2011). Asymptotic and spectral properties of exponentially $\phi$-ergodic Markov processes. Stochastic Process. Appl. 121 1044–1075.
• [21] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
• [22] Kutoyants, Y. A. (1984). Parameter Estimation for Stochastic Processes. Research and Exposition in Mathematics 6. Heldermann, Berlin.
• [23] Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.
• [24] Luschgy, H. and Pagès, G. (2008). Moment estimates for Lévy processes. Electron. Commun. Probab. 13 422–434.
• [25] Mancini, C. (2004). Estimation of the characteristics of the jumps of a general Poisson-diffusion model. Scand. Actuar. J. 1 42–52.
• [26] Maruyama, G. and Tanaka, H. (1959). Ergodic property of $N$-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 13 157–172.
• [27] Masuda, H. (2005). Simple estimators for parametric Markovian trend of ergodic processes based on sampled data. J. Japan Statist. Soc. 35 147–170.
• [28] Masuda, H. (2007). Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps. Stochastic Process. Appl. 117 35–56.
• [29] Masuda, H. (2008). On stability of diffusions with compound-Poisson jumps. Bull. Inform. Cybernet. 40 61–74.
• [30] Masuda, H. (2010). Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein–Uhlenbeck processes. Electron. J. Stat. 4 525–565.
• [31] Masuda, H. (2011). Approximate quadratic estimating function for discretely observed Lévy driven SDEs with application to a noise normality test. RIMS Kôkyûroku 1752 113–131.
• [32] Masuda, H. (2013). Asymptotics for functionals of self-normalized residuals of discretely observed stochastic processes. Stochastic Process. Appl. 123 2752–2778.
• [33] Menaldi, J. L. and Robin, M. (1999). Invariant measure for diffusions with jumps. Appl. Math. Optim. 40 105–140.
• [34] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster–Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25 518–548.
• [35] Ogata, Y. and Inagaki, N. (1977). The weak convergence of the likelihood ratio random fields for Markov observations. Ann. Inst. Statist. Math. 29 165–187.
• [36] Ogihara, T. and Yoshida, N. (2011). Quasi-likelihood analysis for the stochastic differential equation with jumps. Stat. Inference Stoch. Process. 14 189–229.
• [37] Shimizu, Y. (2006). $M$-estimation for discretely observed ergodic diffusion processes with infinitely many jumps. Stat. Inference Stoch. Process. 9 179–225.
• [38] Shimizu, Y. and Yoshida, N. (2006). Estimation of parameters for diffusion processes with jumps from discrete observations. Stat. Inference Stoch. Process. 9 227–277.
• [39] Sieders, A. and Dzhaparidze, K. (1987). A large deviation result for parameter estimators and its application to nonlinear regression analysis. Ann. Statist. 15 1031–1049.
• [40] Sørensen, M. (2008). Efficient estimation for ergodic diffusions sampled at high frequency. Preprint.
• [41] Straumann, D. and Mikosch, T. (2006). Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: A stochastic recurrence equations approach. Ann. Statist. 34 2449–2495.
• [42] Uchida, M. (2010). Contrast-based information criterion for ergodic diffusion processes from discrete observations. Ann. Inst. Statist. Math. 62 161–187.
• [43] Uchida, M. and Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Process. Appl. 122 2885–2924.
• [44] van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge Univ. Press, Cambridge.
• [45] Wang, J. (2008). Criteria for ergodicity of Lévy type operators in dimension one. Stochastic Process. Appl. 118 1909–1928.
• [46] Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss–Newton method. Biometrika 61 439–447.
• [47] Yoshida, N. (1992). Estimation for diffusion processes from discrete observation. J. Multivariate Anal. 41 220–242.
• [48] Yoshida, N. (2011). Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations. Ann. Inst. Statist. Math. 63 431–479.