Fisher's Information for Discretely Sampled Levy Processes

Least squares estimators for discretely observed stochastic processes driven by small Levy noises

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient $\varepsilon \to 0$ and $n \to \infty$ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to t…

Total variation distance for discretely observed Lévy processes: a Gaussian approximation of the small jumps

It is common practice to treat small jumps of Lévy processes as Wiener noise and thus to approximate its marginals by a Gaussian distribution. However, results that allow to quantify the goodness of this approximation according to a given metric are rare. In this paper, we clarify what happens when the chosen metric is the total variation distance. Such a choice is motivated by its statistical interpretation. If the total variation distance between two statistical models converges to zero, then no tests can be constructed to distinguish the two models which are therefore equivalent, statistica…

Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian li…

离散观测Lévy过程的自适应极小极大估计

Adaptive minimax estimation for discretely observed Lévy processes

本文针对离散观测的Lévy过程增量密度非参数估计问题，提出谱估计量，证明其在高低频场景下均达极小极大最优，还提出易实现的数据驱动估计量并通过数值实验验证。

LAN property for an ergodic diffusion with jumps

In this paper, we consider a multidimensional ergodic diffusion with jumps driven by a Brownian motion and a Poisson random measure associated with a pure-jump Lévy process with finite Lévy measure, whose drift coefficient depends on an unknown parameter. Considering the process discretely observed at high frequency, we derive the local asymptotic normality (LAN) property.

Efficient nonparametric inference for discretely observed compound Poisson processes

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as conf…