A sequence of statistical model is Locally Asymptotically Normal (LAN) if, asymptotically, the models converge to a Gaussian shift experiment. Technically, this is if the likelihood ratio processes admit a certain quadratic expansion. The general theory for LAN families provides results both in asymptotic optimality theory and in the behavior of statistical procedures as maximum likelihood estimator and likelihood ratio test. For instance, we are allowed to use the Hajek-Le Cam inequality to defi_ne the asymptotically optimal estimator in problems of parameter estimation. In our paper, we establish LAN for continuous observations from jump-di_ffusion models with time-varying drift and jump intensity, but known volatility. The jump-di_ffusion models are widely used in applications but, in practice, observable data are always discrete so the inference for continuous time models has to be made by discrete samples. In our paper, we show that discrete time high-frequency observations from the same model contain, in an asymptotic and local sense, the same information about the parameters of interest. Basically we show that, the sequence of experiments obtained by continuous time observations and the one obtained by high-frequency data have zero distance in a Le Cam sense. This means that each of the two sequences can be described using the other one and a suitable randomization. More precisely, we provide sufficient conditions on a jump identification mechanism that allows us to construct a central sequence for the continuous time model, using discrete time observations only.
P. Bremaud Point Processes and Queues: Martingale Dynamics
J. Jacod, A. Shiryaev Limit theorems for stochastic processes
U. Küchler, M. Sørensen A Note on Limit Theorems for Multivariate Martingales
Y. Kutoyants Statistical Inference for Ergodic Diffusion Processes
Y. Kwon and C. Lee Strong Feller property and Irreducibility of Diffusions with Jumps
W. Runggaldier Jump-Diffusion Models
K. Sato Lévy Processes and Infinitely Divisible Distributions
Y. Shimizu and N.Yoshida Estimation Parameters for Diffusion Processes with Jumps form Discrete Observations
A. Shiryaev and V. Spokoiny Statistical Experiments and Decisions
Keywords: Local asymptotic normality; Jump-diffusions; High-frequency data; Jump identification
Biography: Irene Gaia Becheri is a PhD student at the Department of Econometrics and Operational Research in Tilburg University. Currently, she is working on inference for stochastic processes, both in continouos and discrete times, and on semiparametric statistics for panel data. She graduated in 2008 cum laude form University of Pisa with a BS and a MSc in Mathematics.