Approximate self-weighted LAD estimation of discretely observed ergodic Ornstein-Uhlenbeck processes

We consider drift estimation of a discretely observed Ornstein-Uhlenbeck process driven by a possibly heavy-tailed symmetric Levy process with positive BlumenthalGetoor activity index β. Under an infill and large-time sampling design, we first establish an asymptotic normality of a self-weighted least absolute deviation estimator with the rate of convergence being √ nh 1−1/β n , where n denotes sample size and hn > 0 the sampling mesh satisfying that hn → 0 and nhn → ∞. This implies that the rate of convergence is determined by the most active part of the driving Levy process; the presence of a driving Wiener part leads to √ nhn, which is familiar in the context of asymptotically efficient estimation of diffusions with compound Poisson jumps, while a pure-jump driving Levy process leads to a faster one. Also discussed is how to construct corresponding asymptotic confidence regions without full specification of the driving Levy process. Second, by means of a polynomial type large deviation inequality we derive convergence of moments of our estimator under additional conditions.

[1]  Shiqing Ling,et al.  Self‐weighted least absolute deviation estimation for infinite variance autoregressive models , 2005 .

[2]  N. Yoshida,et al.  Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property , 2004 .

[3]  N. Yoshida,et al.  Information Criteria in Model Selection for Mixing Processes , 2001 .

[4]  Masanobu Kaneko,et al.  ULTRADISCRETIZATION OF A SOLVABLE TWO-DIMENSIONAL CHAOTIC MAP ASSOCIATED WITH THE HESSE CUBIC CURVE , 2009, 0903.0331.

[5]  M. Sharpe.  Zeroes of Infinitely Divisible Densities , 1969 .

[6]  Yaozhong Hu,et al.  Least squares estimator for Ornstein―Uhlenbeck processes driven by α-stable motions , 2009 .

[7]  Hiroki Masuda.  On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process , 2004 .

[8]  W. Dunsmuir,et al.  Least Absolute Deviation Estimation for Regression with ARMA Errors , 1997 .

[9]  D. Pollard,et al.  Asymptotics for minimisers of convex processes , 2011, 1107.3806.

[10]  Kazufumi Kimoto,et al.  Alpha-determinant cyclic modules and Jacobi polynomials , 2007, 0710.3669.

[11]  R. Koenker,et al.  The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators , 1997 .

[12]  Keith Knight,et al.  Limiting distributions for $L\sb 1$ regression estimators under general conditions , 1998 .

[13]  S. Raible,et al.  Lévy Processes in Finance: Theory, Numerics, and Empirical Facts , 2000 .

[14]  Robert Olkiewicz,et al.  Ornstein-Uhlenbeck-Cauchy process , 1999 .

[15]  Bert Fristedt,et al.  Sample Functions of Stochastic Processes with Stationary, Independent Increments. , 1972 .

[16]  N. Shephard,et al.  Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics , 2001 .

[17]  H. Luschgy.  Local asymptotic mixed normality for semimartingale experiments , 1992 .

[18]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[19]  N. Yoshida,et al.  Asymptotic expansion and information criteria , 2006, SUT Journal of Mathematics.

[20]  J. Rosínski.  Tempering stable processes , 2007 .

[21]  Richard A. Davis,et al.  Maximum likelihood estimation for α-stable autoregressive processes , 2009, 0908.1895.

[22]  SIMPLE ESTIMATORS FOR PARAMETRIC MARKOVIAN TREND OF ERGODIC PROCESSES BASED ON SAMPLED DATA , 2005 .

[23]  J. Doob,et al.  The Brownian Movement and Stochastic Equations , 1942 .

[24]  R. Bhattacharya.  On the functional central limit theorem and the law of the iterated logarithm for Markov processes , 1982 .

[25]  Hiroki Masuda,et al.  Ergodicity and exponential β-mixing bounds for multidimensional diffusions with jumps , 2007 .

[26]  Mtw,et al.  Stochastic flows and stochastic differential equations , 1990 .

[27]  Richard A. Davis,et al.  Maximum likelihood estimation for noncausal autoregressive processes , 1991 .

[28]  N. Yoshida.  Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations , 2011 .

[29]  Richard A. Davis,et al.  M-estimation for autoregressions with infinite variance , 1992 .